Papers

The paper contributes to the problem of finding non-positive invariant Weyl connections on Lie groups. The main result is that in the case of completely solvable 3-dimensional Lie groups only the group SOL admits such connections, confirming the conjecture of Wojtkowski. We conjecture that our method can be extended to the general case of completely solvable Lie groups.

We prove endpoint theorems for one-dimensional bilinear rough singular integrals. Our starting point is a sharp structural characterization of the associated angular multiplier. For every mean-zero $\Omega\in L^1(\mathbb{S}^1)$, the finite-part angular multiplier associated with $T_\Omega$ has bounded variation if and only if the antipodal even part of $\Omega$ belongs to $H^1(\mathbb{S}^1)$. This characterization identifies the precise rotational regularity required in the one-dimensional bilinear setting. It also yields a Stieltjes decomposition compatible with uniform estimates for the bilinear Hilbert transform. We then establish two boundedness criteria under critical kernel assumptions. First, if $\Omega\in L\log L(\mathbb{S}^1)$, then $T_\Omega$ is bounded from $ L^{p_1}(\mathbb{R})\times L^{p_2}(\mathbb{R})\text{to} L^p(\mathbb{R})$ whenever $1<p_1,p_2,p<\infty$ and $ \frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}.$ Moreover, the logarithmic exponent $1$ is optimal within the scale $L(\log L)^A$. Second, at the critical directional index, the same boundedness holds for $\Omega\in\mathcal{K}_{1/2,\beta}(\mathbb{S}^1)$, provided that $\beta>\frac{3}{2}\max\bigl\{p_1,p_1',p_2,p_2'\bigr\}-1.$The two critical kernel classes are incomparable. The $L\log L$ result is obtained by reducing the multiplier to a finite-part angular profile of bounded variation. The directional result follows from endpoint Fourier decay, product wavelet decompositions, and interpolation.

We consider the universal family of smooth genus-$g$ curves over $\mathbb{C}$. We show that for $g\geq4$, every torsor under the relative Jacobian is isomorphic to a connected component of the relative Picard scheme. As a byproduct, we show that $\mathrm{Br}(\mathcal{M}_{3,1})=\mathbb{Z}/2\mathbb{Z}$ over $\mathbb{C}$ and $\mathrm{H}_2(\Gamma_{3,1},\mathbb{Z})=\mathbb{Z}/2\mathbb{Z}$, pinning down the torsion subgroup in \cite[Theorem 1.2]{zbMATH01991000}.

We first propose a notion of $m$-positivity for higher-rank vector bundles, a variant of which reduces to the classical Griffiths positivity when $m=1$. Based on this, we go on to propose a generalisation of the classical Mumford-Takemoto theory of stability by means of a smooth function that we associate with every proper coherent subsheaf ${\cal F}$ of a given holomorphic vector bundle $E$. This places the emphasis on the holomorphic structure and the Hermitian fibre metric of $E$, rather than on numerical invariants of the smooth structure of $E$, making our stability conditions into relative pointwise $m$-positivity properties of $E$ with respect to its proper coherent subsheaves ${\cal F}$. We establish links with Hermite-Einstein geometry, prove that Hermite-Einstein bundles are uniformly semi-stable, study the resulting moduli spaces, and compare the new notions with the classical Mumford-Takemoto (semi-)stability notions.

This work investigates the mean-field stochastic PDEs involving a class of pseudo-monotone kernels. We first study the well-posedness – in both the strong and weak sense – within the variational framework by introducing a notion of measure-dependent pseudo-monotone operators, which generalizes the classical framework due to Brézis. Furthermore, we establish the quantitative dimension-free propagation of chaos within a $p$-uniformly convex Banach space for general infinite-dimensional weakly interacting systems, obtaining convergence rates that are near-optimal in a suitable sense. Our results reveal a new insight: the convergence rate of the mean-field limit is intrinsically governed by the geometry of the underlying solution space, specifically its modulus of convexity. As applications, we study several finite- and infinite-dimensional interacting particle systems arising in machine learning and fluid mechanics, including stochastic Stein variational gradient descent, mean-field Allen-Cahn equations, and Lagrangian-averaged Burgers equations.

Let $N$ be an arbitrary positive integer and let $\lambda=(\lambda_1, \lambda_2, \ldots, \lambda_l)$ be a partition of $N$ of length $l$, i.e., $\sum_{i=1}^l\lambda_i= N$ with parts $\lambda_1\ge \lambda_2\ldots \lambda_l\ge 1$. Define $T(\lambda)$ as the partition of $N$ with parts $l,\lambda_1-1\lambda_2-1,\ldots \lambda_l-1$,ignoring any zeros that might occur. Starting with a partition $\lambda$ of $N$, we describe Bulgarian solitaire by repeatedly applying the shift operation $T$ to obtain the sequence of partitions $$ \lambda, T(\lambda), T^2(\lambda),\ldots . $$ We say a partition $\mu$ of $N$ is $T$-cyclic if $T(\mu) = \mu$ for some $i\ge 1$. In 1982 Brandt [9] characterized all $T$-cyclic partitions for Bulgarian solitaire. Bulgarian solitaire is a dynamical system on integer partition of a positive integer $N$ which converges to a unique fixed point if $N=1+2+\cdots +k$ is a triangular number. In this paper we present a short survey of the game Bulgarian solitaire and several variations of this game.

The use of ratio variables in convex Data Envelopment Analysis (DEA) models has long been recognized as a methodological issue, as ratios may be incompatible with the convexity assumptions underlying standard DEA. However, in this paper, we prove that if all variables are ratio variables sharing a common denominator, then convex combinations of feasible activities are also feasible. Moreover, we establish a result demonstrating the equivalence between DEA models with common-denominator ratio variables under variable returns to scale and DEA models with volume (non-ratio) variables under constant returns to scale. These significant results enable the development of a framework for evaluating the efficiency of the Organisation for Economic Co-operation and Development (OECD) countries based on the results of the Programme for International Student Assessment (PISA) report, using mean performance scores as outputs. In this framework, we give some methodological extensions, such as the incorporation of the index of economic social and cultural status (ESCS) as an input, thereby enabling fairer comparisons with countries with a lower socio-economic level. Furthermore, we introduce different methods for estimating directions of improvement and calculating targets appropriate to the difficulty of improving each performance score. Finally, we review and introduce several novel contributions to emerging methodologies that can complement classical radial and directional models, such as efficient frontier estimation with adaptive constrained enveloping splines (ACES), stochastic chance-constrained models, and fuzzy models. All these methodologies can be used to analyse data from other PISA or similar reports, allowing non-specialists to implement DEA appropriately.

We construct geometric regime-switching diffusions, a class of Markov processes on locally compact stratified Riemannian state spaces. In contrast with classical regime-switching and stochastic hybrid diffusions, the regimes are not external labels, but intrinsic strata of a singular geometric state space. Changes of regime may therefore change dimension, rank, geometry or combinatorial type while the state space maintains its ambient topology. On each stratum the motion is a conservative Feller diffusion, while inter-stratum transitions are specified by state-dependent jump rates and landing kernels along a directed graph. We characterize the process through a martingale problem on a natural stratified core. Under a uniform bound on the total jump rate, we construct a conservative càdlàg strong Markov process by combining the stratumwise diffusions with a Poisson thinning mechanism. Uniqueness is proved using an auxiliary disjoint-union topology and a bounded perturbation argument. Standard Foster–Lyapunov conditions for the extended generator give positive Harris recurrence, uniqueness of the invariant probability measure and, under aperiodicity, \(V\)-uniform geometric ergodicity. The framework is applied to the cone of positive semidefinite covariance matrices, stratified by rank. The resulting process combines fixed-rank covariance diffusions with stochastic rank changes and is \(V\)-uniformly geometrically ergodic.

Let $(X,\omega_0)$ be a compact Kähler manifold of complex dimension $n$, and let $U\Subset X$ have $C^{3,1}$ uniformly strongly pseudoconvex boundary. Consider the signed parallel family $U_t=\{\rho<t\}$, where $\rho$ is a defining function agreeing near $\partial U$ with the signed distance, negative in $U$. Let $u_t$ be the equilibrium envelope with zero obstacle on $X\setminus U_t$, and write $\Sigma_t=\partial U_t$, with outward unit normal $\nu_t$. We prove that the normalized Monge–Ampère energy $e(t)=E_{\omega_0}(u_t)$ belongs to $C^{1,1/2}([-\tau,\tau])$ for some $\tau>0$ and satisfies the Hadamard formula $$ e'(t)=\frac{1}{n+1}\int_{\Sigma_t}\left(-\partial_{\nu_t}u_t\right)^2 d\sigma_{\rho,t}^{\mathrm{tot}}, |t|<\tau. $$ Here $d\sigma_{\rho,t}^{\mathrm{tot}}$ is the interior outward Anzellotti trace of $$ \frac{1}{V} d^c\rho\wedge \sum_{p=0}^{n-1}(p+1) \omega_{u_t}^{p}\wedge\omega_0^{n-1-p}, V=\int_X\omega_0^n, \omega_{u_t}=\omega_0+dd^c u_t. $$ This measure is positive and uniformly comparable to the induced surface measure. The proof combines uniform $C^{1,1}$ estimates up to the boundary from the interior with $1/2$-Hölder stability of the normal derivatives under the normal-flow identifications. We identify the boundary part of the normalized Monge–Ampère measure with the negative outward trace of a divergence-measure flux current. The one-sided weak-* limits of the mixed Bedford–Taylor boundary measures yield the weighted total flux appearing in the variation formula.

The classical Kohn-Temam stress-strain pairing $({\bf A}:E{\bf u})$ for symmetric tensors ${\bf A}$ and ${\bf u}\in BD$ is typically formulated under summability assumptions on the divergence of ${\bf A}$. This excludes stress fields whose divergence has singular surface contributions, as occurs at cracks and material interfaces in continuum mechanics. We define and study stress-strain pairings for bounded symmetric divergence-measure tensor fields. For general ${\bf u}\in BD$, we introduce a slicing pairing $(({\bf A}:E{\bf u}))_\Xi$ for tensor fields satisfying a directional $BV$-type condition with respect to a finite frame $\Xi$. The definition is based on a one-dimensional disintegration strategy, and despite this construction, the new pairing enjoys analogous properties of the usual pairing $({\bf A}:E{\bf u})$, such as the absolutely continuity with respect to $|E{\bf u}|$ and the Gauss-Green formulas. We also identify several situations in which the pairing is independent of the choice of frame $\Xi$, including the relevant case in which the stress field ${\bf A}$ belongs to $BV$. While a distributional stress-strain pairing can be defined naturally for bounded $BD$ functions, it cannot be extended to the unbounded setting, since the truncation techniques available in $BV$ fail in $BD$. The slicing pairing is consistent with the distributional one whenever the latter is defined, while being more general even for bounded ${\bf u}$. Indeed, its existence does not require the compatibility condition $|{\rm Div}\,{\bf A}|(S_{{\bf u}}\setminus J_{\bf u})=0$ which is necessary for the distributional definition. This allows the treatment of stress fields interacting with diffuse micro-cracking.

The author is interested in regions surrounded by hypersurfaces and natural smooth maps onto them respecting the canonical projections of the unit spheres and so-called special generic maps and moment maps, more generally. We consider situations where these regions are foliated via 1-dimensional families of functions and their zero sets (smoothly). We including the author are also interested in explicit and nice functions obtained by composing the canonical projections and their topological or combinatorial properties. This is of singularity theory of differentiable maps and applications to differential topology and various real geometry. Explicitly, here, as a new challenge, we discuss the 1st derivative of such a function and critical sets of this.

In this work we present a novel discrete collocation method for the numerical solution of Fredholm integral equations of the second kind in the space of continuous functions equipped with the uniform norm. The method is based on a rational interpolation scheme recently developed within the general framework of reproducing kernel Hilbert spaces. This rational approximation has no real poles, interpolates the target function at arbitrary Jacobi nodes and exhibits uniformly bounded Lebesgue constants. Moreover, it converges uniformly for all continuous functions at a rate at least equal to that of the best uniform polynomial approximation. These interesting properties are inherited by the resulting numerical method, for which stability, convergence and good conditioning are established under minimal assumptions on the integral kernel. A series of numerical experiments confirm the theoretical findings and indicate that, in the presence of particularly challenging kernels, the proposed approach provides a robust and effective alternative to Nyström-type methods.

For a class of non-autonomous linear SPDEs, we establish the equivalence among asymptotic regularization, weak observability, and approximate null controllability for the associated deterministic control systems. This equivalence provides a deterministic control-theoretic characterization of stochastic smoothing and offers a systematic approach to studying SPDEs driven by spatially localized noise. We further establish a criterion for semilinear SPDEs based on weak observability of the linearized equations. Our approach combines methods from PDE control theory with Malliavin calculus. As applications, we consider the stochastic Oseen equation, non-autonomous uniformly parabolic equations, and the parabolic Sine–Gordon equation, all driven by finite-dimensional, spatially localized white-in-time noise.

The randomness, efficiency (power and variability), and desirable allocation proportions are important components for evaluating a response-adaptive design in clinical trials and conflicted demands in applications. The aim of this paper is to provide designs dealing with these dilemmas. We first give a general framework for efficient response-adaptive randomization procedures that attain the Cramér-Rao lower bounds of the allocation variances for any desired allocation proportions. The general framework is flexible for us to define new families of efficient designs with good properties for both two and multiple-treatment clinical trials. We also prove that, among all response-adaptive randomization procedures with the same limit allocation proportions, the selection biases and entropies as measures of the randomness of the designs have their optimal values. Basing on the theory on efficiency and randomness, we propose a new family of doubly adaptive biased coin designs for multi-treatment clinical trials that can target any allocation proportion and are asymptotically best in terms both the randomness and efficiency so that their randomness is asymptotic optimal and asymptotic allocation variance attains the Cramér-Rao lower bound. Theoretical properties, including the strong consistency, the asymptotic normality, and the functional central limit theorem for both the sample allocation proportions and the estimators of the distribution parameters, are developed by using the technique of Gaussian approximation and Gaussian comparing theorems.

QBism interprets quantum theory as a normative discipline for an agent's probability assignments and their revision across possible experience. This paper develops a logical formalization of that picture. A well-formed core datum consists of an admissible prior space, a finite family of actual measurements, Born kernels, and update kernels. For each such datum, we introduce a guarded dynamic language for histories and posterior states and prove a global reduction theorem. We next consider effectively semialgebraic data over an effectively presented real closed field. For data in this class, we translate the fragment without dynamic operators into first-order formulas in the corresponding language of ordered rings, thereby reducing validity to first-order reasoning over real closed fields. Together, the reduction and first-order translation yield a sound and complete recursive calculus and a decision procedure for validity. Finally, assuming a symmetric informationally complete (SIC) reference measurement, we show that quantum theory in finite dimensions realizes the framework through SIC coordinates, POVMs, and quantum instruments. We also prove that the corresponding SIC image satisfies the standard qplex geometry conditions, namely the consistency bounds and the lower polar condition, and that under explicit coefficient field hypotheses the resulting quantum datum is effectively semialgebraic.

We provide a complete characterization of the bounded, compact, and Hilbert-Schmidt class weighted composition operators acting on weighted Fock spaces, extending the classical Fock space characterization due to Le [11]. An estimate for the essential norm of these operators is also provided. Our approach relies on the asymptotic behavior of the Mittag-Leffler function. As applications, we establish explicit criteria for weighted composition operators with exponential weights and recover corresponding results for composition operators.

Reconfigurable intelligent surface (RIS)-assisted over-the-air federated learning (OTA-FL) enables efficient distributed intelligence but suffers from time-varying channels, imperfect channel state information (CSI), and strong user heterogeneity, which jointly degrade aggregation accuracy and cause severe model update cancellation. To address these issues, we propose a unified framework for joint channel estimation and dynamics-aware user grouping in RIS-assisted OTA-FL systems, enabling reliable learning under imperfect CSI and heterogeneous dynamics. The framework integrates gated recurrent unit (GRU) for temporal modeling to capture time-varying CSI evolution, OTA-based federated aggregation with personalization, and RIS-aware physical-layer optimization in a closed loop. In addition, we design a dynamics-aware grouping strategy based on long-term path-loss and short-term channel dynamics to reduce inter-user conflicts under heterogeneous conditions. Simulation results show that the proposed method achieves substantial gains in CSI estimation accuracy and OTA aggregation performance in low-pilot and high-mobility regimes, while improving convergence speed and robustness under strong user heterogeneity.

Denote by ${\mathcal A}$ the Grassmann algebra with a countable number of generators, by ${\mathcal A}_{\overline 0}$, ${\mathcal A}_{\overline 1}$ its even and odd parts. We consider supergroups as groups over $\mathcal A$. Consider the Neveu–Schwarz Lie superalgebra $\mathfrak{ns}= \mathfrak{ns}_{\overline 0}\oplus \mathfrak{ns}_{\overline 1}$. Consider its Grassmannization $\mathfrak{ns}(\mathcal{A}):= (\mathfrak{ns}_{\overline 0}\otimes {\mathcal A}_{\overline 0}) \oplus (\mathfrak{ns}_{\overline 1}\otimes {\mathcal A}_{\overline 1})$ and the corresponding supergroup $\mathrm{NS}(\mathcal {A})$. We describe this group explicitly in different ways. Next, we define a complexification $\Gamma({\mathcal A})$ of $\mathrm{NS}({\mathcal A})$. It is a semigroup, whose elements are superannuli of dimension $1|1$ equipped with contact structures; the multiplication is gluing of such superannuli. Under some inequalities for parameters, we show that unitary representations of $\mathrm{NS}({\mathcal A})$ admit extensions to representations of $\Gamma(\mathcal {A})$. We do not assume that the reader has prior knowledge of Lie superalgebras and supergroups.

We investigate the normal form method in piecewise-smooth monodromic systems to distinguish center from focus and determine the order of focus as well as degenerate Hopf bifurcations. For the normal form of piecewise-smooth monodromic systems given in previous publications in the sense of topologically equivalence, we prove the algebraic equivalence between the Lyapunov constants of the original piecewise-smooth system and the coefficient series of its normal form. With this algebraic equivalence, we prefer to compute the normal form coefficient series to avoid cumbersome integrals of trigonometric functions in the computation of Lyapunov constants, and find the relations of foci of the piecewise-smooth system and its subsystems: among the orders and among the stabilities. Moreover, we design algorithms for computing the normal form coefficient series and combine them with the obtained stability relation to construct stabilizing switching signals for state-dependent switched nonlinear systems, one basic problem in the control theory.

Let $G$ be a connected reductive group over a $p$-adic field $F$, $U$ the unipotent radical of a minimal parabolic subgroup, $\psi$ a depth-zero non-degenerate character of $U(F)$, and $I$ an Iwahori subgroup of $G(F)$. We show that, as a module over the Iwahori-Hecke algebra ${H}$, the space of $I$-fixed vectors in the Gelfand-Graev representation $\mathrm{ind}_U^G\psi$ is isomorphic to ${H} \otimes_{{H}_{W_0}} \mathrm{sgn}$. Here $\mathrm{sgn}$ is the sign representation of the finite Hecke subalgebra ${H}_{W_0}$ attached to the relative Weyl group. This extends the theorem of Chan-Savin from split groups to all connected reductive groups.

In this work, we develop a unified matrix-level asymptotic calculus for plug-in non-parametric maximum likelihood estimators in finite Markov models. Starting from the asymptotic distribution of the estimated transition matrix, the limiting object is kept in its natural matrix form as a Gaussian random matrix, while the corresponding row-wise vector representation remains immediately available. The main point is that the stochastic constraints of the transition matrix need not be removed by a minimal parametrization: they are carried by the tangent directions and by the covariance structure of the limiting Gaussian matrix, whereas the relevant differentials are computed directly in matrix spaces. A single stochastic calculus theorem gives first-order limit distributions, finite-order developments for sufficiently differentiable functionals, and analytic expansions when the functional is analytic. This provides a common source for asymptotic formulas for matrix powers, stationary characteristics, finite-dimensional curves of Markov characteristics, additive-functional variances, entropy-type quantities and reliability indicators. The resulting covariance operators lead directly to confidence intervals, confidence regions, simultaneous finite-dimensional bands and Wald-type tests. Since the derivations are expressed through matrix products and Kronecker representations rather than coordinate-wise calculations, the method also gives substantial simplifications and, in many cases, computational gains. The second-order terms identify curvature corrections of smooth functionals and provide refined approximations whenever higher-order information is useful.

We study Heisenberg uniqueness for the positive hyperbola branch and shifted lattice crosses in the supercritical regime $q=\alpha\gamma>1$. We resolve the infinite-dimensionality clause of the arbitrary-shift problem posed by Giri and Manna: for arbitrary shifts on both arms, the normalized pre-annihilator is infinite-dimensional. More precisely, every $v\in BV((1,q))$ has a global $BV$ pre-annihilating extension; the extension is unique unless both twisting phases are trivial, in which case its ambiguity is one-dimensional. The proof reduces the annihilation conditions to a graph equation for a twisted Perron–Frobenius operator and combines a phase-uniform Lasota–Yorke estimate with peripheral spectral rigidity. We also give an exact operator-theoretic normal form for the entire $L^1$ pre-annihilator in terms of the maximal convergence domain of the associated Green series. Writing $Q$ for the twisted product and $A$ for the forcing operator, we show that $Q$ has the closed unit disk as its spectrum on $L^1((0,1))$, that $\Ran(I-Q)$ is not closed, and that, outside a countable set of algebraic values of $q>1$, the operator $\sum_{j=0}^{N-1}Q^jA:L^1((1,q))\to L^1((0,1))$ has norm $2N$ for every $N\ge1$.

Graph k-CUT problems include many important variants whose objectives combine cut value, volume, and cardinality terms in different ways. Most existing algorithms are designed for individual formulations, which limits their transferability across related models. In this paper, we organize a broad family of graph partitioning problems into two classes, MaxGCP and MinGCP, according to their optimization orientation and balance-related structure. Based on this classification, we propose a unified Parallel Evolutionary Algorithm Framework (PEAF). This framework combines structure-inheriting crossover operators, a hierarchical mutation mechanism based on the Multiple Mutation Heuristic (MMH) and the Auxiliary Cut Mutation Heuristic (ACMH), and a diversity-preserving selection strategy. Extensive experiments on G-set with k \in\{2, 3, 4, 5\} show that PEAF-ACMH consistently outperforms Gurobi on nine representative k-CUT problems. For MaxGCP, PEAF-ACMH improves several best-known solutions for Max-k-Cut with k \geq 3, and through numerical bounds derived from its relation to Max-k-Cut, verifies the high quality of the obtained solutions for Judicious-k-Partition and AntiCheeger-k-Cut. The results further indicate that Judicious-k-Partition usually yields more balanced partitions than AntiCheeger-k-Cut. For MinGCP, theoretical and computational comparisons show that Cheeger-k-Cut and Sparsest-k-Cut produce more balanced partitions than Normalized-k-Cut and Ratio-k-Cut, respectively. PEAF-ACMH also obtains highly similar partitions for Min-k-Cut and MinMax-k-Cut within short running times, providing numerical evidence for their structural affinity. These results demonstrate that PEAF is both an effective unified solver and a useful tool for revealing structural properties of graph k-CUT models.

The dynamic fracture behaviour of two moving collinear Griffith cracks in an initially stressed dry sandy medium subjected to concentrated crack-face loading and moving punch pressure is investigated. A moving coordinate system transforms the transient problem into a steady state formulation, while the effects of initial stress and sandiness are incorporated into the governing equations. Fourier integral transforms are employed to obtain the characteristic equation and the transformed traction displacement relations. The crack face, symmetry, and outer surface conditions reduce the problem to coupled Cauchy type singular integral equations. For a sufficiently thick strip, an asymptotic kernel reduction is developed, and the resulting equations are solved analytically using the finite Hilbert transform. Closed form expressions are derived for the crack density functions, Mode I stress intensity factors at the inner and outer crack tips, and the crack opening displacement. Several limiting cases are recovered from the general formulation, providing analytical consistency checks. The results demonstrate the combined influence of crack speed, crack geometry, initial stress, sandiness, and moving punch loading on crack tip intensification and crack opening. The proposed analytical framework provides useful insights for fracture assessment and the design of transportation infrastructure, geotechnical systems, underground excavations, and other engineering structures involving dry sandy media subjected to moving loads.

Binary relations between finite sets N and X, families of subsets, and lattice homomorphisms from P(N) to P(X) are three faces of the same object. This self-contained monograph, merging and substantially expanding the author's ICTP preprint IC/97/180 (1997) and a 2026 companion note, enumerates every combinatorial type (arbitrary, injective, surjective, on either side), both raw and up to the actions of the symmetric groups S_N, S_X, and S_N x S_X, via the Cauchy-Frobenius-Burnside lemma. The generating functions of all counts are then determined, revealing e^{z+w+zw}, Bell numbers, Euler's partition product, and the quasi-polynomiality of the hardest, two-sided column of the table. All prerequisite tools – elementary counting, group actions, Stirling and Bell numbers, formal power series, and Mobius inversion on the Boolean lattice – are developed from first principles, making the book accessible after a first course in algebra and combinatorics. A documented correction to Theorem B of the 1997 preprint is included; all formulas have been verified by symbolic computation and brute-force enumeration.

Let $k$ be a positive integer. Jackson and Thomassen conjectured in 1989 that there exists an integer function $f(k)$ such that every $f(k)$-strong digraph admits a spanning $k$-strong oriented subdigraph. They even conjectured that one can take $f(k)=2k$ [Ann. N. Y. Acad. Sci. 555 (1989) 402-412]. Already the existence of $f(2)$ is open for general digraphs. Thomassen proved that $f(2)=4$ for symmetric digraphs. For general $k$, the existence of $f(k)$ was only known for locally semicomplete digraphs and quasi-transitive digraphs. Guo proved that every ${(3k-2)}$-strong locally semicomplete digraph contains a spanning $k$-strong local tournament [Discrete Appl. Math. 79 (1997) 119–125]. One can deduce from Guo's result that we have $f(k)\leq 3k-2$ for quasi-transitive digraphs. In this paper, we prove the existence of $f(k)$ for two subclasses of the semicomplete multipartite digraphs, namely extended semicomplete digraphs and semicomplete split digraphs. We prove that every $(4k+1)$-strong extended semicomplete digraph contains a spanning $k$-strong oriented subdigraph and every $5k$-strong semicomplete split digraph contains a spanning $k$-strong oriented subdigraph. The first result implies that for the large class of digraphs which can be obtained from some semicomplete digraph $S$ on at least 3 vertices by substituting arbitrary digraphs for each vertex of $S$ we also have $f(k)\leq 4k+1$.

Every finite subquandle of a spherical quandle is accounted for by dihedral quandles and by quandles defined on root systems.

A lowest-order mixed finite element method is developed for the strain gradient elasticity (SGE) model in arbitrary dimensions. We take the physically meaningful third-order double stress tensor $\boldsymbol{\Phi}:=\iota^2\operatorname{grad}\boldsymbol{\sigma}(\boldsymbol{u})\in\mathbb{S}\otimes\mathbb{R}^d$ as a primary variable and derive a distributional mixed formulation. The double stress is approximated by an $\mathbb{S}\otimes\mathbb{R}^d$-valued extension of the lowest-order Raviart–Thomas element, while the displacement is approximated by the vector-valued linear Crouzeix–Raviart element. Thus, the method avoids both high-degree bubble enrichment and a Nitsche-type treatment of the higher-order boundary condition. We establish parameter-robust discrete stability, an optimal first-order error estimate for fixed parameters, and a complementary parameter-uniform $\mathcal{O}(\iota^{1/2}+h)$ error estimate with constants independent of both the size parameter $\iota$ and the Lamé coefficient $\lambda$. In the boundary-layer regime $\iota^{1/2}\lesssim h$, the latter retains a first-order convergence rate in $h$. We also develop a local quadratic post-processing and a hybridized formulation. Numerical experiments in two and three dimensions support the theoretical results.

Wireless communication networks are evolving toward extremely large antenna arrays, millimeter-wave and terahertz bands, and dense heterogeneous deployments, all of which increase channel dimensionality and make channel acquisition increasingly costly. Channel knowledge map (CKM) establishes a mapping from geographical locations to channel characteristics, providing location-specific prior information to reduce the overhead of channel acquisition. Most existing CKM research, however, has focused on quasi-static propagation features shaped by quasi-static environmental structures such as buildings and terrain, leaving unaddressed the time-varying channel component introduced by dynamic scatterers, terminal attitude changes, and radio-frequency (RF) impairments. This article presents a new concept of dynamic CKM as a middle layer that links quasi-static environmental priors to physical-layer signal processing by providing time-evolving channel representations. We first introduce the fundamentals of dynamic CKM, clarifying its relationship with the quasi-static CKM and the physical layer. We then survey representative construction methods and discuss how dynamic CKM can support pilot design, interference suppression, and integrated sensing and communications. Finally, we outline key open research directions in the co-design of dynamic CKM construction and physical-layer signal processing. These discussions offer an architectural perspective on the role of dynamic CKM in emerging 6G systems.

We give an introduction to the grammatical calculus, which originated from the umbral calculus of Rota. Context-free grammars arise in computer science. Differential operators can be viewed as the production rules of context-free grammars. By the properties of differential operators, we may perform the grammatical calculus on a rigorous basis, for the purposes of deriving combinatorial identities and computing generating functions. A grammatical labeling serves as a bridge between a combinatorial structure and the grammar. A grammar may be instrumental in constructing a bijection between two combinatorial structures sharing the same grammar. Such a bijection is called a grammar assisted bijection. Many combinatorial structures admit recursive constructions that are context-free in some sense, such as those corresponding to the Eulerian polynomials, Bell polynomials, the Ramanujan polynomials and the Narayana polynomials.