Papers

We propose the use of a covariance-aware detection method for single-radio-frequency (RF) multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) systems. This detection approach enhances bit error rate (BER) performance as compared to conventional detectors. The approach is motivated by the observation that when using reconfigurable antennas (such as electronically steerable parasitic array radiators (ESPARs)) to create a single-RF MIMO system, an additional model error arising from the reconfigurable antennas is introduced. These modeling errors produce colored noise and an irreducible BER (error floor) at high signal-to-noise ratios (SNRs). Simulation results, using ESPAR as an example, validate our error floor analysis and demonstrate that the use of the proposed enhanced detection method can effectively address the error floor and improve the BER at high transmit SNRs.

In this paper, we study the class \Sigma_{H}^{k}(p) of sense-preserving univalent harmonic mappings in the unit disk \mathbb{D} that possess a simple pole at p\in[0,1) and admit a k-quasiconformal extension to the extended complex plane for k\in[0,1). In 2024, Bhowmik and Satpati established an area theorem and derived a sufficient condition for the k-quasiconformal extension of harmonic mappings belonging to \Sigma_{H}^{k}(p) without logarithmic terms. Motivated by their work, we investigate the corresponding problem when a logarithmic singularity is present. Our main contributions are two-fold: we first prove a generalized area theorem for all mappings in \Sigma_{H}^{k}(p); we then obtain a sufficient condition for sense-preserving univalent harmonic mappings in \mathbb{D} to admit explicit k-quasiconformal extensions. These results extend the aforementioned work to the setting where logarithmic singularities are allowed.

Johnson graph are a family of graphs that play an important role in the theory of constant-weight codes, extremal combinatorics, and combinatorial geometry. We study signed analogues of classical Johnson graphs, denoted by $J_\pm(n,k)$, whose vertices are vectors of the form $\pm e_{i_1}\pm\cdots\pm e_{i_k}$, where two vertices are adjacent whenever their dot product equals $k-1$. We are particularly interested in maximum independent sets in the case $k=4$. An example of such an independent set in $J_\pm(n,4)$, which we call classical, is obtained by lifting an arbitrary optimal $(n,4,4)$-code. Such independent sets naturally define kissing arrangements in ${\mathbb R}^n$. We develop an algorithm that is practical for computing all maximum independent sets in $J_\pm(n,4)$ up to signed permutations for $n\le 12$, $n\ne 11$. In addition to obtaining complete lists, we provide structural characterizations of all types of maximum independent sets in these dimensions, excluding $n=5$ and $n=11$. Our most striking results concern the case $n=12$. We identify $1579$ non-isomorphic maximum independent sets in $J_\pm(12,4)$, all corresponding to non-isometric kissing arrangements of size $840$ in ${\mathbb R}^{12}$. Structurally, $1575$ of these independent sets arise from three different constructions, the rest are liftings of one of four $(12,4,4)$-codes. To our knowledge, this is the first dimension in which such a large diversity of potentially optimal kissing arrangements has been observed. Beyond this finite range, we prove that for $n\equiv 2$ or $4 \pmod 6$, every maximum independent set arises from a Steiner quadruple system. We also obtain a characterization of the so-called nontrivially self-compatible codes, namely optimal $(n,4,4)$-codes from which non-classical maximum independent sets can be constructed.

A pair $(x,y)$ of vectors in a Banach space $\mathcal{Y}$ is said to be a triangle equality attaining (or TEA) pair if $\|x+ y\|=\|x\|+\|y\|,$ and a parallel pair if $\|x+\lambda y\|=\|x\|+\|y\|$ holds for some unimodular scalar $\lambda.$ In this article, we explore bounded linear maps $T:\ell_p\to \mathcal{Y}$ preserving parallel and TEA pairs. For $p\in(1,\infty)$ all linear maps trivially preserve parallel pairs. We prove that for $p=\infty,$ if $\ker(T)\neq \{0\},$ then $T$ preserves parallel pairs if and only if $rank(T)\leq 1$. %In particular, $T:\ell_p\to \ell_1$ preserves parallel pairs if and only if $rank(T)\leq 1,$ and TEA pairs if and only if $T=0$. In particular, $T:\ell_\infty\to \ell_1$ preserves parallel (resp. TEA) pairs if and only if $rank(T)\leq 1$ (resp. $T=0$). Analogous characterizations hold if $T$ is defined from $\ell_\infty^n,$ except when $n=2$ and the field is real. In this specific setting, we further characterize such maps $T:\ell_\infty^2\to \ell_1.$ \\ Focusing on $p=1$, we establish a necessary condition for the preservation of parallel pairs. Specifically, we characterize invertible parallel pair preservers $T:\ell_1^n\to\ell_\infty^n$, as well as the general class of such maps $T:\ell_1^2 \to \ell_\infty^m,$ revealing the intricate structure inherent to these mappings. Furthermore, we prove that $(0\neq)~T:\ell_1\to \mathcal{Y}$ preserves TEA pairs if and only if $\Lambda=\{i\in \mathbb{N}:Te_i\neq 0\}$ is singleton, where $\mathcal{Y}$ is either strictly convex or $\ell_\infty^m$ over the complex field. Finally we characterize the TEA pair preservers $T:\ell_1^2\to \ell_\infty^m$ over the real field.

This paper investigates an equality-coupled distributed resource allocation problem with smooth general convex local objective functions. A discrete-time residual-aware dynamic event-triggered algorithm is proposed over time-varying switching undirected graphs. Unlike existing event-triggered resource allocation algorithms that rely on strong convexity, the proposed method establishes convergence for general convex costs without using strong monotonicity or contraction arguments. The key idea is to co-design a resource-allocation search recursion with a dynamic triggering rule that incorporates both local gradient-estimation errors and local gradient-disagreement residuals. The resulting triggering mechanism reduces unnecessary communication and generates a summable error bound, which is embedded into a Mirror-EXTRA-type Lyapunov analysis. Under suitable step-size conditions, the proposed algorithm is proved to converge to an optimal solution. Further, when the local objective functions are strongly convex, a linear convergence result is established. Numerical simulations and comparative tests with related event-triggered methods verify the effectiveness and communication efficiency of the proposed algorithm.

A Hessian-recovery-based C0 finite element framework is proposed for second-order elliptic equations in non-divergence form. The construction is based on a direct approximation of the strong non-divergence operator: the Hessian D2u is replaced by a recovered Hessian Hhuh, so that A : D2u is approximated by A : Hhuh. The resulting discretizations include a nodal formulation and a Galerkin-type formulation for general Lagrange finite element spaces, as well as a biorthogonal Petrov-Galerkin formulation for linear elements. The analysis focuses on the recovered nodal matrix and identifies two verifiable algebraic solvability mechanisms. The first is a globally monotone regime leading to a discrete maximum principle, and the second is a localized Schur-complement criterion for sign-violating rows. A uniform inverse bound and a condition-number estimate are derived in the globally monotone case. Residual consistency estimates are obtained from the Hessian recovery error. In the globally monotone regime, these estimates combine with the uniform inverse bound to give a nodal L-error estimate for the nodal formulation. Numerical experiments with nonsmooth and discontinuous coefficients support the predicted algebraic diagnostics and show the accuracy of the proposed recovered-residual discretizations. A Monge-Ampere type test further illustrates the use of the recovered Hessian in a Newton iteration for a fully nonlinear problem.

In this paper, we consider triangle-free graphs with diameter 2. If a triangle-free graph $G$ with diameter 2 is not isomorphic to a star, then the radius of $G$ is also 2, where such a graph is also called a $2$-self-centered graph. Shekarriz et al. [A characterization for 2-self-centered graphs, Discuss. Math. Graph Theory 38 (2018), 27–37.] gave a characterization of 2-self-centered graphs. However, there is a slight flaw in their characterization. Thus, in this paper, we modify it and prove an accurate characterization of those graphs. Furthermore, by using our characterization, we prove some results concerning the chromatic number of triangle-free graphs with diameter 2.

Neural operators learn mappings between infinite-dimensional function spaces and provide a data-driven surrogate modeling paradigm for parametric partial differential equations (PDEs). Existing architectures typically obtain expressivity by parameterizing integral kernels in prescribed transform domains or by applying attention-like interactions over discretized spatial points. While these approaches have achieved substantial progress, they often face a persistent trade-off among physical interpretability, nonlocal spatial communication, mesh scalability, and computational cost. We propose a Light-inspired neural operator(LiNO), an operator-learning architecture whose latent evolution is decomposed into three mechanisms motivated by elementary light transport: reflection, refraction, and scattering. Reflection and refraction act as adaptive pointwise transformations in latent feature space, enabling local feature reorientation and anisotropic modulation, whereas scattering performs input-dependent nonlocal propagation over the physical domain. We first formulate scattering as a normalized pairwise kernel with relative positional bias, and then develop an efficient scattering variant that replaces explicit pairwise interactions with positive-feature global propagation and a local diffusion branch, reducing the dominant spatial complexity from quadratic to linear. This yields a structured neural operator that separates local feature modulation from global spatial communication while retaining a modular and interpretable latent evolution.

Pinching antennas are characterized by their capability to create strong line-of-sight connections and realize multi-antenna systems in a flexible manner. Existing works have demonstrated the significant potential of pinching antennas for physical layer design. The aim of this paper is to investigate how pinching antennas can be used to reshape the architecture of future networks. In particular, this paper is motivated by the key advantage of pinching antennas, which is to reconfigure the physical boundaries of wireless cells, and focuses on the impact of pinching antennas on traffic offloading. The models for traffic offloading and pinching antenna transmission are presented first. Then, two traffic offloading strategies are developed based on whether an offloading user releases its bandwidth in its original cell. An overall transmit power minimization problem is formulated, where the optimal solutions for the transmit powers and antenna locations are obtained. The presented simulation results demonstrate that the use of pinching antennas can efficiently support traffic offloading, yield low energy consumption, and achieve balanced cell resource utilization.

In this paper, a special chain representation of real numbers on a fixed interval is considered, where the elements of the expansion can take only one of two possible values. For this encoding system, a problem in metric number theory and dynamical systems is solved, which is a direct analogue of the classical Gauss problem for simple continued fractions. Specifically, the asymptotic behavior of the Lebesgue measure for a special class of sets is investigated. These sets are formed by those numbers for which the remainder (or infinite tail) of their chain expansion, after discarding the first few elements, is strictly less than a predetermined value.

Efforts have been made to extend Cassini's identity (also known as Simson's identity) to the k-step or k-bonacci numbers for decades. These efforts have lacked both completeness of result and simplicity of proof, and this question remains open and relevant. In this note, we offer a definitive solution as well as the generalization of both Catalan's and Vajda's identities.

This letter studies robust set-point regulation of scalar second-order nonlinear uncertain systems using a classical PID controller with constant gains. The scalar second-order model provides a minimal prototype for nonlinear mechanical and electromechanical dynamics, while its velocity-dependent term captures uncertainties such as physical damping and friction. For a positive velocity-derivative bound, existing Lyapunov sufficient conditions certify fixed-gain PID parameter regions that remain separated from the boundary associated with the necessary condition obtained from the worst-case linear model. To reduce this conservatism, this letter proposes an endpoint-balanced quadratic-plus-integral Lyapunov certificate. The key idea is to choose the quadratic cross-term coefficient so that the mixed-term penalty is balanced at the two endpoints of the admissible effective-damping interval before extracting the scalar PID inequality. The resulting condition guarantees global asymptotic regulation for the full derivative-bounded uncertainty class. When the velocity-derivative bound is positive, the proposed condition certifies a fixed-gain PID region that strictly contains those certified by Zhao–Guo and Zhang–Guo. When this bound is zero, the corresponding boundary coincides with that necessary boundary. At the level of Lyapunov analysis, the construction reduces the uniform mixed-term penalty over the entire effective-damping interval.

Automatic-differentiation-based inverse analysis methods, including physics-informed neural networks (PINNs) and differentiable programming, have recently shown great promise due to their ability to compute accurate gradients and convergence efficiency. However, their applicability to falling weight deflectometer (FWD) backcalculation remains unexplored. This study critically evaluates PINN-based inverse analysis for a multilayer pavement system and investigates differentiable finite element method (DiffFEM) as an alternative based on a synthetic benchmark. The standard PINN does not recover layer moduli because of the sharp domain discontinuities inherent to layered pavement systems. Although we use an extended PINN with domain decomposition (XPINN), which shows better performance on discontinuous domains, its performance remains highly sensitive to loss weighting and network architecture, and degrades under measurement noise. By contrast, DiffFEM consistently achieves more accurate, stable, and computationally efficient inversion results. These results show and quantify the expected advantage of DiffFEM, which enforces the governing physics as a hard constraint in the solver, over PINN-based approaches, where the governing physics is imposed as a soft constraint through the loss function. More broadly, the findings suggest that the choice between PINN- and DiffFEM-based inverse analysis needs careful consideration, with DiffFEM offering practical advantages when an efficient and robust differentiable forward solver is available.

We introduce and study the model category of flexible $2$-categories, which is Quillen equivalent to Lack's model category of $2$-categories, but enjoys several excellent properties not shared by the latter. In particular, every object of this model category is cofibrant, it is a monoidal model category with respect to its cartesian closed structure, and its full subcategory of fibrant objects is equivalent to the category of bicategories and normal pseudofunctors.

The Membership Problem for Pedigree Polytope (M3P) asks, given $X\in\mathbb{Q}^{\binom{n}{3}}$, whether $X\in\mathrm{conv}(P_n)$, where $P_n$ is the set of all pedigrees. A pedigree is a structured encoding of a Hamiltonian cycle construction in $K_n$. We establish that M3P is solvable in strongly polynomial time via a recursively constructed layered network $(N_k, R_k, \mu)$ and a multicommodity flow problem MCF$(k)$. The necessary and sufficient condition for membership established is that the optimal total flow in MCF$(n-1)$ equals the maximum possible flow $z_{\max}$. The complexity analysis, grounded in Tardos's strongly polynomial algorithm for combinatorial linear programs (1986), shows that this condition can be checked in strongly polynomial time in the dimension of the matrix involved. By sufficiency, this implies M3P~$\in$~P. Since the Symmetric Travelling Salesman Problem (STSP) reduces to M3P via the Multistage Insertion (MI) formulation (Arthanari 1983), STSP is solvable in polynomial time, and the P vs.NP question is resolved. The proofs leading to this result are fully machine-verified in Lean~4/Mathlib4, with zero unresolved sorrys in the main proof chain. The main contribution is the Lean~4 machine verification of all proofs in the main chain, resulting in theorem p\_equals\_np: P = NP. The Lean~4 formal verification covers the sufficiency of MCF(n-1) for membership in $\mathrm{conv}(P_n)$, and the P = NP chain via Maurras (2002), Grötschel–Lovász–Schrijver (1988), Cook (1971), and Karp (1972). The complete lean project (36 Lean~4 files, 2968/2968 build targets clean) is available at https://github.com/TiruArt/Pedigree-Polytopes-Lean4.

In this paper we prove a conjectured equivalence of categories, showing that the S-dual of $\mathrm{SO}_{2n}\times \mathrm{Sp}_{2n}$ acting on $\mathbb{C}_+^{2n}\otimes \mathbb{C}_-^{2n}$ is equal to $\mathrm{SO}_{2n+1}\times \mathrm{SO}_{2n}\circlearrowright T^*\mathrm{SO}_{2n+1}$. This result is a particular case of a non-polarized version of the (local) relative Langlands duality of Ben Zvi, Sakellaridis and Venkatesh. Similar results for the pairs $(\mathrm{SO}_{2n+1}, \mathrm{Sp}_{2n})$ and $(\mathrm{GL}_n, \mathrm{GL}_m)$ were proved by Braverman, Finkelberg, Kazhdan and Travkin and by Fu respectively, whereas the converse result was proved by Braverman, Finkelberg, and Travkin. As a consequence of our main result, we prove that Langlands functoriality of the Derived Satake isomorphism for the pair $\mathrm{Sp}_{2n},\mathrm{SO}_{2n}$ is given by the theta correspondence. Our approach works (with appropriate modifications) in the general even orthosymplectic case of $\mathfrak{osp}(2m|2n)$.

In this paper, we study the noncommutative fractional symmetric Sobolev spaces on noncommutative torus. We prove noncommutative distributional fractional Sobolev inequality and as its application, we obtain Sobolev embeddings. In order to obtain these results, we first prove a noncommutative version of the famous O'Neil inequality for the convolution. As a first application of our main results, we obtain a Cwikel-Solomyak-type estimate. As an another application, we show a $L_2$-time decay for the mild solution of the Cauchy problem for the diffusion equation in this noncommutative setting. When $\theta=0,$ our results recover many known results on Sobolev embedding on the torus.

The stochastic incompressible Navier-Stokes equations on $\TT^3$, completed by the fluctuation-dissipation noise, have a Fokker-Planck generator that decomposes into a self-adjoint Ornstein-Uhlenbeck (dissipative) part and an antisymmetric (convective) part. We prove two results about this generator. First, the logarithmic Sobolev inequality holds with the same optimal constant as the pure Ornstein-Uhlenbeck operator, $c_\mathrm{LSI} = \nu\lambda_1$ (where $\nu$ is the viscosity and $\lambda_1$ is the smallest nonzero eigenvalue of the Laplacian on $\TT^3$), independent of the number of retained Fourier modes. Second, the full semigroup is hypercontractive with the same rate as the Ornstein-Uhlenbeck semigroup. Both results follow from a single structural property: the convective generator is antisymmetric in $L^2(P_\mathrm{eq})$ (where $P_\mathrm{eq}$ is the Gibbs measure), and therefore contributes nothing to the Dirichlet form or the $L^q$ norm evolution. The antisymmetry is a consequence of two properties of the incompressible Navier-Stokes nonlinearity: energy conservation and phase-space volume preservation (the Liouville property). These are the same properties that underpin the fluctuation-dissipation theorem for the nonlinear Navier-Stokes equations.

We conjecture that on any compact balanced manifold $(M, \omega_B)$ with $c_{1}(M)=0$, the pluriclosed flow admits long-time solutions $\omega_{t}$ for every initial pluriclosed metric, and that $\omega_{t}$ converges smoothly to a Kähler metric as $t \to \infty$. We verify that this phenomenon occurs when $M$ is a compact quotient of a Lie group by a discrete subgroup, the background metric $\omega_{B}$ is invariant with vanishing Chern–Ricci form, and the initial metric $\omega_{0}$ is invariant. In particular, this provides new evidences for the Fino-Vezzoni conjecture.

The partial Petrial polynomial of a ribbon graph, introduced by Gross, Mansour and Tucker, enumerates partial Petrials by Euler genus. Recently, Deng, Jin and Yan defined an analogue for grafts and showed that it can be expressed as a rank-generating function of an adjacency matrix. In this paper we first prove a recurrence relation that reduces the partial Petrial polynomial of a graph with respect to an arbitrary edge, expressing it as a sum of three terms involving graphs obtained by local complementation and edge pivoting. This recurrence extends the known leaf-reduction formula to vertices of any positive degree. Second, using this recurrence we compare the lowest and highest degree coefficients of the polynomial. We prove that the lowest coefficient is always at most the highest coefficient, and that equality holds if and only if the graph has no edges.

We investigate the convergence properties of the family of operators $$ \mathcal{T}_{\bf N} f(x)=\sum_{j=N_1}^{N_2} v_j\Bigl( e^{-a_{j+1}(-\Delta_k)^s}f(x) - e^{-a_j(-\Delta_k)^s}f(x) \Bigr), \qquad x\in\mathbb{R}^d, $$ where $\{e^{-t(-\Delta_k)^s}\}_{t>0}$ denotes the fractional heat semigroup generated by the Dunkl Laplacian $\Delta_k$. Here ${\bf N}=(N_1,N_2)\in\mathbb{Z}^2$, $N_1<N_2$, the coefficients $\{v_j\}_{j\in\mathbb{Z}}$ form a bounded sequence of real numbers, and $\{a_j\}_{j\in\mathbb{Z}}$ is a monotone increasing sequence of nonnegative reals. The primary objective of this work is to establish boundedness results for these differential transform operators on weighted $L^p(\mathbb{R}^d,d\mu_k)$ spaces as well as on Dunkl $\mathrm{BMO}(\mathbb{R}^d)$ spaces. We also establish analogous boundedness properties for the associated maximal operator $$ \mathcal{T}^* f(x)=\sup_{\bf N} |\mathcal{T}_{\bf N} f(x)| $$ and study the pointwise convergence of the corresponding series. In addition, we prove that, for compactly supported functions, the maximal differential transform operator $\mathcal{T}^*$ exhibits local behaviour comparable to that of classical singular integral operators.

Let $X \subset \mathbb{R}^2$ be a closed definable set of dimension at most $1$, and let $h : X \to \mathbb{R}^2$ be a definable continuous injective map. In this paper, we establish necessary and sufficient combinatorial conditions, formulated in terms of cyclic orders at topological singular points and orientations of Jordan curves, for $h$ to admit a definable homeomorphic extension to the whole plane.

In this paper, we establish new connections between lattice graphs and metric grids, providing a unified framework for the study of singular limit problems and Gagliardo–Nirenberg type inequalities on lattice graphs. The main technical ingredients are restriction and extension estimates, which enable us to compare variational problems posed on lattice graphs, metric grids and \(\mathbb R^d\). As applications, we first prove that extensions of action ($2<p<2^*$) and energy ($2<p<2+\frac{4}{d}$) ground states of the nonlinear Schrödinger (NLS) equation on $d$-dimensional lattice graphs converge strongly in $H^1(\R^d)$ to the corresponding ground states on $\R^d$ as the edge length tends to zero. As a by-product of the arguments developed for the singular limit problem on lattice graphs, we obtain multiplicity results for fixed-mass critical points of the energy functional on lattice graphs. Furthermore, employing a strategy analogous to that used in the singular limit analysis, we investigate the optimal constants of Gagliardo-Nirenberg type inequalities on lattice graphs for $2<p<2^*$. Beyond the classical subcritical framework, we also study the singular limit of action ground states in the Sobolev supercritical regime ($d \geq 3$ and $p>2^*$), the singular limit of energy ground states in the mass-supercritical regime ($p>2+\frac{4}{d}$) on lattice graphs, and the optimal constants in Gagliardo-Nirenberg type inequalities in the Sobolev critical case $d \geq 3$ and $p=2^*$ on lattice graphs. Notably, we settle an open problem posed by Dovetta [Adv. Math. 444 (2024), 109633] by establishing a new Gagliardo-Nirenberg type inequality.

We construct $h$- and $p$-robust, degree-preserving space decompositions and additive Schwarz preconditioners for variable-degree $hp$ finite element discretizations of conforming reaction-diffusion and fitted-interface problems. On conforming simplicial meshes, an owner-selected Falk–Winther bubble transform gives $L^2$- and $H^1$-stable components with constants independent of the mesh size, the local polynomial degrees, and the degree distribution. Minimal-degree owners preserve arbitrary variable-degree spaces with $p_K\ge1$, while coefficient-adapted owners yield weighted estimates under local chain conditions. Combined with a weighted continuous piecewise affine extraction, this gives $hp$-uniform Schwarz preconditioners for conforming reaction-diffusion problems with locally comparable coefficients, and a coefficient-weighted conforming variant in the uniform-degree case. For three-dimensional fitted-interface problems, we use a symmetric Nitsche discretization on a tetrahedral mesh fitted to a piecewise planar interface. Surface jump components are lifted into the side selected by the penalty scaling using patch-level $p$-robust trace liftings. The conforming remainder is decomposed by the low-order extraction and a weighted one-sided bubble transform. Grouping the resulting components by vertices yields a practical vertex-patch Schwarz preconditioner whose condition number is independent of the mesh size, local polynomial degrees, diffusion contrast, and coefficient magnitudes under a common-degree condition on interface-touching tetrahedra. Numerical experiments for pure diffusion problems support the theory and suggest robustness beyond the common-degree assumption.

The hypersequent calculus $\text{G\L}\forall$, an analytic Gentzen-style proof system of first-order Łukasiewicz logic, and its approximate completeness have been extensively studied. In this paper, we prove the approximate strong completeness of $\text{G\L}\forall$ by a labelled tableau method. Then we introduce a sequent-level cut rule (s-Cut) and show the approximate strong completeness of $\text{G\L}\forall+(\text{s-Cut})$. As applications, we establish the compactness of approximate $[0, 1]$-consequence, a variant of Gentzen's mid-sequent theorem of $\text{G\L}\forall$ and an approximate Herbrand's theorem of first-order Łukasiewicz logic.

Spectral sequences are a key tool for computing the K-theory of a crossed product C$^*$-algebra. However, the impact of a group homomorphism $\Omega\colon G \to H$ on such a spectral sequence was unknown until quite recently, even when $G = \mathbb Z^\ell$, $H = \mathbb Z^{k}.$ Recent work [Mil25] of the fourth-named author in the complex case establishes that ABC spectral sequences are functorial with respect to group homomorphisms. In this paper, we obtain the analogous result for real K-theory and for united K-theory. Specifically, we first show that the ABC spectral sequence approximates KO$_*(G \ltimes_r A)$ with the group homology H$_p(G;KO_q(A))$ when $G$ is a torsion-free discrete group satisfying the Baum–Connes conjecture with coefficients in $A$. Then, for a homomorphism $\Omega \colon G \to H$ of such groups with amenable kernel, and a real $H$-C$^*$-algebra $A$, we show moreover that the map in K-theory induced by the $*$-homomorphism $G \ltimes_r A \to H \ltimes_r A$ is approximated by the natural map in group homology.

We study the spatial propagation of super-Brownian motion on $\mathbb{R}^d$ with general critical or subcritical (spatially dependent) branching mechanisms. Under local spatial lower bounds satisfying a Keller-Osserman type integrability condition, we establish a quantitative upper bound for the short-time probability that the support exits a prescribed neighborhood of its initial support. The estimate has a Gaussian-tail form and is obtained through weighted occupation times, Feynman-Kac representations, singular elliptic boundary blow-up estimates, and mild comparison arguments for log-Laplace equations. As an application, we derive the compact support property directly for spatially dependent branching mechanisms satisfying suitable local lower bounds. This yields a sufficient compact-support criterion expressed in terms of the inverse Keller integral. In particular, for spatially dependent super-Brownian motions with stable branching, we generalize the compact support results of Engländer-Pinsky and Ren.

A split graph is a graph where the vertices are a disjoint union of a complete part $C=\{x_i,\ldots,x_n\}$ and a stable part $S=\{y_1,\ldots,y_m\}$. We will determine the Betti numbers of the edge ring of all split graphs, in particular show that the only nonzero Betti numbers are $\beta_{0,0}$ and $\beta_{i,i+1}$, $i>0$. The Betti numbers only depend on the multiset of the number of neighbors in $S$ the $x_i$'s have. Singh and Verma have earlier determined the Betti numbers for complete split graphs (where all $y_i$ are neighbors to all $x_j$), and for "nearly complete" split graphs (where all $y_i$ are neighbors to all $x_j$, except that $y_i$ is not a neighbor to $x_i$ for $i=1,\ldots,\min\{m,n\}$). We also determine which split graphs that have Cohen-Macaulay edge ring.

Let $\Lambda \subset\mathbb{R}^3$ be a region admitting a partition into $n$ compact, connected subregions $\Lambda_1,\dots,\Lambda_n$, each with smooth boundary. Consider a vector field $B$ on $\Lambda$ where $B|_{\Lambda_i}$ is smooth, divergence free, and tangent to $\partial \Lambda_i$ for all $i$. We show that the multi-region relaxed magnetohydrodynamics (MRxMHD) equilibrium equations are necessary and sufficient conditions for $ B $ and a metric to yield a stationary point of the magnetic energy under appropriate constraints. We constrain the pressure, relative helicity, and magnetic flux of $B$ through all smooth surfaces in $\Lambda_i$ whose boundary lies on $\partial \Lambda_i$. We identify a previously overlooked gauge condition. A definition for relative helicity is introduced, its gauge invariance is proved, and the existence of a gauge where relative helicity reduces to conventional helicity is demonstrated. In the case of a single region an additional condition is introduced that is sufficient to ensure a critical point of the magnetic energy is also a minimiser.

Let $C$ be a nonempty closed and convex subset of a uniformly smooth and uniformly convex real Banach space $E$ with dual space $E^{*}$. We introduce a viscosity-Halpern hybrid projection scheme for approximating a common element of the fixed point set of a countable family of generalized nonexpansive-type mappings, the solution sets of countably many variational inequality problems, and the solution sets of countably many equilibrium problems. The method combines a viscosity perturbation generated by a contraction, a Halpern anchor term, equilibrium and variational inequality resolvent steps, and a shrinking generalized projection step. Under monotonicity, continuity, closedness and NST-type assumptions, we prove strong convergence of the generated sequence to the generalized projection of the initial point onto the common solution set. We also give a generalized-projection variational characterization of the selected limit, residual convergence, Hilbert-space specializations, and examples showing that the full countable problem cannot, in general, be recovered from finite truncations.