In this paper we give estimate of $|S'_{f}(0)|$, where $S_{f}(z)= \left(\frac{f''(z)}{f'(z)}\right)'-\frac{1}{2}\left(\frac{f''(z)}{f'(z)}\right)^{2}$ is the Schwartian derivative, and $f$ belongs to different classes of functions univalent in the open unit disc $\ID$.

In this article, we analyze how (projective and injective) complexity, curvature, and complete intersection dimension behave under linkage of modules and ideals. Let $R$ be a Gorenstein local ring. Consider a Gorenstein perfect ideal $\mathfrak{a}$ (e.g., $\mathfrak{a}$ is generated by an $R$-regular sequence). Let $M$ and $N$ be two Cohen-Macaulay $R$-modules linked by $\mathfrak{a}$. We prove that $\mathrm{cx}_R(M)= \mathrm{inj\,cx}_R(N)$ and $\mathrm{curv}_R(M)= \mathrm{inj\,curv}_R(N)$. In particular, when $R$ is complete intersection, $\mathrm{cx}_R(M)= \mathrm{cx}_R(N)$ and $\mathrm{curv}_R(M)= \mathrm{curv}_R(N)$. Furthermore, we show that $\mathrm{pd}_R(M)= \mathrm{pd}_R(N)$ and $\operatorname{CI-dim}_R(M)= \operatorname{CI-dim}_R(N)$. If any of these dimensions is finite, it is equal to $\mathrm{ht}(\mathfrak{a})$. Similar results are obtained for linkage of ideals. All these results highly extend a classical result of Peskine and Szpiro in many directions. We construct several examples that complement our results. These also show how properties like `integrally closed', `$\mathfrak{m}$-full' and `Burch' behave under linkage of ideals.

Bondal's conjecture in Poisson geometry gives lower bounds on the degeneracy loci of Poisson Fano manifolds, where the rank of the Poisson structure drops. By work of several authors, it was previously known to hold for Fano manifolds of dimension at most four. We give the first proof of this conjecture for Fano manifolds of dimension five, and partial results for Fano manifolds of all odd dimensions. The proof uses: (i) an algebraic integrability criterion for codimension-one foliations on weak Fano manifolds, extending a previous result of the first author; (ii) the "modular residues" of Poisson structures introduced by Gualtieri and the third author; and (iii) a cohomological constraint on invariant subvarieties for Pfaff fields, extending earlier results of Esteves--Kleiman to the case in which the Pfaff distribution on the subvariety admits a closed strongly directed positive current.

Implication algebras were introduced by Abbott as algebraic models of the operation of Boolean implication in the classical propositional calculus. In this work, we study additional operators and constants on bounded implication algebras by introducing monadic and cylindric implication algebras. It is demonstrated that the category $\mathbf{MIA}$ of monadic implication algebras is isomorphic to the category $\mathbf{MBA}$ of monadic Boolean algebras and moreover, that the category $\mathbf{CIA}$ of $I$-dimensional cylindric implication algebras is isomorphic to the category $\mathbf{CBA}$ of $I$-dimensional cylindric Boolean algebras. As an application of the obtained categorical isomorphisms, we provide spectral duality results for $I$-dimensional cylindric implication algebras along the lines of Bezhanishvili and Holliday's spectral duality for Boolean algebras combined with McDonald's extension of their duality to monadic and $I$-dimensional cylindric Boolean algebras.

Stone demonstrated that the category $\mathbf{DLATT_{0,1}}$ of bounded distributive lattices is dually equivalent to the category $\mathbf{Spec}$ of spectral spaces and Priestley showed that $\mathbf{DLatt_{0,1}}$ is dually equivalent to the category $\mathbf{Priest}$ of Priestley spaces so that $\mathbf{Spec}$ is equivalent $\mathbf{Priest}$. Cornish strengthened this by showing that $\mathbf{Spec}$ and $\mathbf{Priest}$ are in fact isomorphic. In this study, we investigate the duality theory of various lattice expansions of certain bounded distributive lattice-ordered algebras, known as De Morgan algebras. In particular we obtain spectral duality results for the category $\mathbf{S4DM}$ of De Morgan algebras equipped with a closure operator, which we call S4 De Morgan algebras, as well as for the category $\mathbf{DMGrp}$ of De Morgan groupoids. This is achieved by an appropriate adaptation of Bimbó's Priestley-style duality for general De Morgan algebras together with Urquhart's Priestley-style duality for relevance algebras under the isomorphism between $\mathbf{Priest}$ and $\mathbf{Spec}$.

We investigate a recursive variant of the classical Frank-Wolfe algorithm for minimizing a convex differentiable function over a convex compact domain. Unlike the traditional setting, we assume that both the problem domain and the objective function are initially unknown and must be learned from data. To address this, we integrate estimators into the optimization process, allowing the algorithm to iteratively refine approximations of the problem domain and the objective function. Our approach maintains the projection-free nature of the classical Frank-Wolfe algorithm while adapting to the uncertainty inherent in data-driven settings. We establish convergence guarantees for the recursive method, showing that the optimization error scales with the accuracy of the learned estimators. Two experiments support our theoretical findings, demonstrating that the proposed method achieves convergence behavior comparable to that of the classical Frank-Wolfe algorithm under exact knowledge of the problem domain and objective function, while offering significant computational savings.

We study integrability of generalized almost complex structures on the six-dimensional sphere $\mathbb{S}^6$. Two notions of integrability are considered: integrability with respect to brackets determined by an affine connection $[,]_\nabla$ (in particular the Levi-Civita connection), and the Courant integrability for strong generalized almost complex structures. After recalling the necessary background on the generalized tangent bundle and on spherical combinations of the canonical generalized structures determined by an almost Hermitian triple $(J,g,ω)$, we derive local coordinate criteria for $[,]_\nabla$-integrability of weak generalized structures. Applying these formulae to the nearly Kähler structure on S^6 induced by the octonionic product, we prove that no nontrivial spherical combinations $J=aJ_{1,J} + bJ_g + cJ_ω$ with smooth coefficients such that $a^2+b^2+c^2=1$ (except $J_g$) is integrable with respect to $[,]_{\nabla^{LC}}$. We then turn to Courant integrability: we give sufficient local conditions for Courant integrability of strong generalized almost complex structures, prove a gluing result for local Courant algebroids and b-field transforms, and use it to exhibit obstruction results characterizing the impossibility of constructing, via certain gluing procedures, a Courant integrable strong generalized almost complex structures on $\mathbb{S}^6$.

Let $\mathcal{I}$ be a $σ$-ideal on the Cantor space $2^ω$. A set $X\subseteq 2^ω$ is called everywhere $\mathcal{I}$ if for any $S\in [ω]^ω$ the set $X\!\restriction\! S\in\mathcal{I}(2^S).$ We will discuss relations between such families in the class of perfect and Borel sets.

We prove an upper bound for the number of subgroups of an abelian $p$-group, with constants that are sharp on homocyclic blocks. If $A_λ$ has type $λ$, the main exponent is $η(λ)=\sum_i\lfloor(λ'_i)^2/4\rfloor$, where the parts of $λ'$ are the column heights of the Ferrers diagram of $λ$. The leading coefficient is the number of central choices in the blocks of columns of odd height. The proof starts from Delsarte's formula, separates the blocks by column height, and estimates each homocyclic block by successive Durfee squares. We also give three applications: a comparison theorem for $p$-groups of nilpotency class less than $p$, diagonal summatory estimates with the shape fixed, and formulae for leading terms at fixed rank for finite abelian $p$-groups.

This paper proposes a stochastic control framework for the operation of electric-vehicle (EV) charging stations equipped with on-site photovoltaic (PV) generation and battery storage. To preserve scalability for large fleets, vehicles are aggregated into a finite number of classes according to their residual charging demand, yielding a compact state description and avoiding vehicle-level optimization. The charging-station dynamics are modeled in discrete time and capture stochastic arrivals, charging-induced class transitions, stochastic departures, PV generation, battery operation, and power exchange with the grid. Based on the corresponding expectation model, we formulate a finite-horizon smart-charging problem that jointly optimizes class-wise charging actions and energy-management variables to balance electricity-purchase cost and transitions toward lower residual-demand classes. The resulting stage problem is a linear program, solved in shrinking-horizon form and implemented online after integer discretization of the first charging action. We also derive a robust counterpart that preserves feasibility under interval uncertainty on the first moments of arrivals, departures, and PV generation. To validate scalability and robustness, we conduct an extensive simulation campaign across nine configurations, combining three real-world electricity price patterns with three EV-arrival profiles. Numerical results show that, at comparable but slightly lower service levels, the proposed controller reduces the cost per kWh by up to 17.5% relative to a service-greedy First-In-First-Out baseline; the total daily cost decreases even more substantially because the controller avoids economically unattractive charging. Overall, the results demonstrate a tunable Pareto trade-off between economic efficiency and charging-service quality for large-scale charging hubs.

A novel detection method based on maximum-likelihood (ML) detection leveraging Mahalanobis distance is proposed for single-radio-frequency (RF) multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) systems. It can enhance bit error rate (BER) performance and is based on 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 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 our proposed enhanced detection method can effectively address the error floor and reduce the BER at high transmit SNRs.

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 \emph{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 \emph{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 $\|x+λy\|=\|x\|+\|y\|$ holds for some unimodular scalar $λ.$ 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 $Λ=\{i\in \mathbb{N}:Te_i\neq 0\}$ 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.

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.

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 $\mathfrak{osp}(2m|2n)$.

We conjecture that on any compact balanced manifold $(M, ω_B)$ with $c_{1}(M)=0$, the pluriclosed flow admits long-time solutions $ω_{t}$ for every initial pluriclosed metric, and that $ω_{t}$ 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 $ω_{B}$ is invariant with vanishing Chern--Ricci form, and the initial metric $ω_{0}$ 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 $$ \mathcal{T}_{\bf N} f(x)=\sum_{j=N_1}^{N_2} v_j\Bigl( e^{-a_{j+1}(-Δ_k)^s}f(x) - e^{-a_j(-Δ_k)^s}f(x) \Bigr), \qquad x\in\mathbb{R}^d, $$ where $\{e^{-t(-Δ_k)^s}\}_{t>0}$ denotes the fractional heat semigroup generated by the Dunkl Laplacian $Δ_k$. 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 reals. The primary objective of this work is to establish boundedness results for these differential transform operators on weighted $L^p(\mathbb{R}^d,dμ_k)$ 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. As applications, we first show 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 $H^1(\mathbb{R}^d)$ to the corresponding ground states on $\mathbb{R}^d$ as the edge length of lattice graphs tends to zero. As a by-product of the arguments developed for the singular limit problem of lattice graphs, we obtain multiplicity results for fixed-mass critical points of energy functional on lattice graphs. Furthermore, employing a strategy analogous to the analysis of the singular limit problem of lattice graphs, we investigate the optimal constants of Gagliardo--Nirenberg type inequalities on lattice graphs for $2<p<2^*$. A distinctive feature of this paper is that, going beyond the classical subcritical framework, we also establish novel results on the singular limit problem of action ($d \geq 3$ and $p>2^*$) and energy ($p>2+\frac{4}{d}$) ground states on lattice graphs, and on the optimal constants of Gagliardo--Nirenberg type inequalities for $d \geq 3$ and $p=2^*$ on lattice graphs, thereby substantially extending the existing literature. Notably, we settle an open problem posed by Dovetta [Adv. Math. 444 (2024), 109633] by establishing a new Gagliardo--Nirenberg type inequality.

Spectral sequences are a key tool for computing the K-theory of a crossed product C$^*$-algebra. However, the impact of a group homomorphism $Ω\colon G \to H$ 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 $Ω\colon G \to H$ 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 branching mechanisms. Under a Keller-Osserman type integrability condition on the spatially homogeneous branching mechanism, 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 comparison principles for nonlinear PDEs. As an application, we derive the compact support property for super-Brownian motions started from compactly supported finite measures. We further extend the argument to spatially dependent branching mechanisms by imposing suitable local lower bounds given by spatially homogeneous mechanisms. This yields a sufficient compact-support criterion expressed in terms of the inverse Keller integral. In particular, our results extend those of Sheu for super-Brownian motions with spatially constant branching mechanisms, and those of Engländer-Pinsky and Ren for super-Brownian motions with stable-like spatially dependent branching mechanisms.

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 $β_{0,0}$ and $β_{i,i+1}$, $i>0$. 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 $C$ 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.