Papers

Let $q=2^m$, $Q=2^k$, and $1\leq k\leq m-1$. We characterize complete permutation polynomials (CPPs) over $\mathbb{F}_{q^2}$ of the form \[ f(x)=c_0x^{Q+1}+c_1x^{Q+q}+c_2x^{qQ+1} +c_3x^{q(Q+1)},\qquad c_i\in\mathbb{F}_{q^2}. \] We prove that no such CPP exists when $k>1$, and recover the known characterization in the cubic case $k=1$. This completes the classification throughout the stated exponent range. The proof uses the known permutation classification to reduce completeness to linear perturbations of product and monomial models. The required nonpermutation results follow from direct elementary arguments based on the quadratic structure over $\mathbb{F}_2$.

This paper investigates a multi-agent pursuit problem in which a group of pursuers seeks to intercept one or more evaders within a fixed operational time horizon. The motion of both pursuers and evaders is described by first-order linear differential equations, and the control inputs available to the agents are subject to Gronwall-type constraints. Pursuit is considered successful when at least one pursuer reaches the position of each evader at a finite time within the prescribed duration. Within this framework, we construct permissible strategies for the pursuers and derive sufficient conditions under which interception is guaranteed despite the evaders' counteractions. To support the theoretical results, we present numerical examples that demonstrate the effectiveness of the proposed strategies and confirm the analytical conditions for successful pursuit.

We develop a variational–Markov construction of detailed-balance master-equation discretizations for Fokker–Planck equations directly from their energy–dissipation laws. Rather than discretizing the differential operator, we represent mass transfer between neighboring grid points by two directional jump rates. On each edge, the discrete energy–dissipation law determines the net flux, while detailed balance fixes the rate ratio; a local reconstruction of edge quantities from neighboring grid-point values then uniquely determines both rates. A logarithmic-mean reconstruction recovers the classical Scharfetter–Gummel/Wang–Peskin–Elston rates, while alternative reconstructions yield other reversible schemes. The resulting semi-discrete systems conserve mass, preserve nonnegativity, satisfy detailed balance, and dissipate a discrete free energy. The same edgewise construction extends to state-dependent and degenerate mobilities, nonlocal interaction energies and higher-dimensional problems. The corresponding implicit and semi-implicit fully discrete schemes are linear, mass-conservative, positivity-preserving, and energy-stable, with a time-step restriction required only for nonlocal interaction energies whose kernel is not negative semidefinite. Numerical experiments confirm the expected spatial convergence rates of all detailed-balance schemes and verify their equilibrium accuracy, positivity preservation, and free-energy decay with discontinuous potentials, saturation, nonlocal interactions, and two-dimensional problems.

We study the set $\Gamma$ of graphs of closed, densely defined operators in a Hilbert space $H$, regarded as a subset of the Grassmann manifold $P(H\times H)$ of orthogonal projections in $H\times H$. We show that the subset $\Gamma^b$ of graphs of bounded operators is the open unit ball of $P(H\times H)$ centered at the graph of the zero operator $P_0$ (which projects onto $H\times\{0\}$). This ball is diffeomorphic to $B(H)$ via the map $T\mapsto P_T$ ($=$ the projection onto the graph ${Gr(T)}$ of $T$). We show that graphs of unbounded closed operators lie at the boundary of $\Gamma$. We also study the existence and characteristics of minimal geodesics of $P(H\times H)$ joining two graphs $Gr(A)$, $Gr(B)$. If $A,B$ are selfadjoint, such a geodesic always exists, and we construct explicitly a distinguished exponent using the five-space decomposition associated to the pair of subspaces $Gr(A)$, $Gr(B)$. An explicit low-dimensional example shows that the geodesic joining two graphs need not remain inside $\Gamma$, i.e., does not consist entirely of graphs. We also relate graphs of compact operators to the restricted Grassmannian, and study the problem of common complements for pairs $Gr(S)$, $Gr(T)$, giving positive results when one operator is bounded or under lower boundedness conditions.

We prove that on any closed manifold there exists a C1 open and dense subset of star flows for which the periodic orbits grow at most expo- nentially. We also explain how our methods apply to the sectional-hyperbolic attractors including the Lorenz polynomial equation.

Noise modulation (NoiseMod) encodes information in the variance of a noise-like waveform. Its noncoherent structure enables low-complexity links, but co-channel users are challenging because interference affects the same variance statistic used for detection. We study fluid antenna multiple access (FAMA) for two-level NoiseMod with independent random bits from each interferer, producing two possible nonzero interference variances. We adopt exact finite-sample maximum-likelihood energy detection from prior NoiseMod/TNM work. We derive the exact single-port moment-generating function and moments of the random-bit aggregate interference, including heterogeneous user powers, and prove that the port minimizing conditional BEP is $\arg\max_k H_k/(\sigma_w^2+J_k)$, where $H_k$ is desired-link power and $J_k$ is instantaneous bit-state-aware interference variance. For independent fading ports, the common interferer-bit vector couples per-port decision metrics; conditioning on the number of high-state interferers yields an exact binomial-mixture order-statistic representation and avoids the generally invalid rule $F_{Z_{\max}}=F_Z^{N_p}$. For spatially correlated operation, an integer-$\mu$ $\kappa$-$\mu$ benchmark uses full pairwise Jakes correlation in scattered fields, with channel-averaged BEP evaluated by conditional Monte Carlo. A standard-error-aware load study shows increased admissible co-channel users as spatial degrees of freedom grow. Finally, a finite-sample variance-domain sensing diagnostic quantifies the oracle gap under frozen channels and interferer states, providing a lower-bound warning on fast-FAMA sensing difficulty rather than a deployable acquisition protocol.

This paper establishes a comprehensive combinatorial and arithmetic framework for mixed Stirling and mixed Bell numbers, bridging partition structures, Touchard polynomials, and prime-power congruences. Furthermore, we develop to $p$-adic valuation theory, proving prime-power Touchard congruences and higher-order modulo-$p^2$ refinements that generalize classical arithmetic properties of combinatorial sequences.

We obtain a slight improvement to the trilinear bound of Guth-Zahl by application of Wang-Zahl's resolution of 3-D Kakeya to tubes inside a thickened neighborhood of a variety. This is combined with the planebrush argument of Katz and Zahl to obtain the new bound of 3.073.

Let $\theta$ be an ordinal and let $(N_\alpha)_{\alpha\leq\theta}$ be a continuous increasing tower of closed normal subgroups of a Hausdorff topological group $G$, with $N_0=\{e\}$. Write $\rho G$ for the Raikov completion of $G$, and put $K_\alpha=\overline{N_\alpha}^{\rho G}$ and $H_\alpha=GK_\alpha$. The successive and limit differences of the groups $H_\alpha$ mark the first stages at which points of $\rho G$ become old quotient points. For every nonzero ordinal $\theta$ and every set $S\subseteq(0,\theta]$, there is a Hausdorff pseudocompact Boolean group with a continuous tower of closed normal subgroups such that every member of the tower is pseudocompact and the nonempty strata occur exactly at the stages in $S$. The terminal quotient can simultaneously be chosen isomorphic to $\mathbb Z/2\mathbb Z$, and for every $\delta<\theta$ the successive factor $N_{\delta+1}/N_\delta$ is compact exactly when $\delta+1\notin S$. Thus a tower may have a single nonempty stratum at a limit stage even when every successive factor is compact. For a closed normal subgroup $N$ of $G$, let $\widehat q:\rho G\to\rho(G/N)$ extend the quotient homomorphism and put $K=\overline N^{\rho G}$. Over $gN\in G/N$, the fiber in $\rho G\setminus G$ is $g(K\setminus N)$, homeomorphic to $\rho N\setminus N$, while for $y\in\widehat q(\rho G)\setminus(G/N)$ the entire $\widehat q$-fiber lies in $\rho G\setminus G$ and is homeomorphic to $\rho N$. For precompact kernels this gives a canonical decomposition, and nested quotients give the two-stage identity underlying the transfinite filtration. Further results concern pseudocompact Raikov remainders and include countable Abelian examples.

Moustrou, Riener, and Verdure related the dominance order on partitions to Specht ideal and variety inclusion for $S_n$; we previously extended this to direct products of symmetric groups. Here we take a first step toward the imprimitive complex reflection group $G(r,1,n)=(\mathbb{Z}/r\mathbb{Z})\wr S_n$. The correct geometric invariant of a point $x\in K^n$ is a pair $(\mu,z)$: a classical partition $\mu$ on the $r$-th powers of the nonzero coordinates, together with the number $z$ of zero coordinates. We construct a $G(r,1,n)$-invariant ideal $I_{(\mu,z)}$ for each such pair, and show that a dominance order on $(\mu,z)$, with $z$ taking strict priority over $\mu$, exactly characterizes both ideal inclusion and reverse variety inclusion. We settle a monomial criterion in three regimes, recover the classical $B_n$-Specht ideal theory at $r=2$ when $z=0$, and establish radicality of $I_{(\mu,z)}$ at the two extremes of the $z=0$ level. Four computational examples, checked exhaustively for group invariance, illustrate the construction throughout.

We construct a uniformly Kreiss bounded operator on a separable Hilbert space which is not strongly Cesàro bounded. This gives a negative answer to a question of Cohen, Cuny, Eisner and Lin.

Let $K$ be a finite pure ridge-connected $d$-dimensional simplicial complex with at least two facets, and let $m_K(\lambda)$ denote the multiplicity of a positive eigenvalue $\lambda$ of its upper Laplacian $L_{d-1}^{\mathrm{up}}(K)$. We obtain upper bounds for $m_K(\lambda)$ in terms of the basic combinatorial parameters $f_d(K)$ and $f_{d-1}(K)$, the numbers of facets and ridges, $p_d(K)$, the number of pendant facets, and $q_d(K)$, the number of quasi-pendant ridges. The quantity $\theta_d(K)=d f_d(K)-f_{d-1}(K)+1$ arises as the cyclomatic number of a natural reduced facet–ridge incidence graph. We prove \[ m_K(\lambda)\le 2\theta_d(K)+p_d(K), \qquad m_K(\lambda)\le 2\theta_d(K)+q_d(K)\quad(\lambda\ne d). \] We also characterize the complexes for which equality is attained in these bounds. Finally, for every finite pure $d$-dimensional simplicial complex, we prove the sharp inequality $m_K(d)\ge p_d(K)-q_d(K)$, which is a higher-dimensional analogue of Faria's inequality. For $d=1$, our results reduce to the corresponding multiplicity bounds for graph Laplacians.

We establish higher regularity of the regular free boundary for non-negative solutions of the Kolmogorov obstacle problem \[ \mathcal K u=\chi_{\{u>0\}}, \qquad \mathcal K=\Delta_x+x\cdot\nabla_y-\partial_t. \] Assume that the contact set satisfies a quantitative thickness condition at a free-boundary point. We prove that, in a neighborhood of that point, the free boundary is a non-characteristic \(C_{\mathcal K}^{1,\beta}\) hypersurface for some \(\beta\in(0,1)\). Equivalently, its inward non-degenerate normal is Hölder continuous with respect to the intrinsic Kolmogorov distance, and the deviation of the graph from its tangent hyperplane is of order \(O(r^{1+\beta})\) at intrinsic scale \(r\). The proof combines the half-space blow-up and differentiability theory at regular points with boundary Harnack estimates in asymptotically cylindrical intrinsic Lipschitz domains. A principal difficulty is that the diffusion derivatives determining the normal do not satisfy the homogeneous Kolmogorov equation. We introduce a \(\mathcal K\)-harmonic-replacement argument that converts the commutator sources into an \(O(r)\) error under intrinsic blow-up and yields a contraction estimate for the oscillations of derivative quotients. This gives Hölder continuity of the non-degenerate normal. In addition, a boundary decay estimate for the transport derivative $Yu=(x\cdot\nabla_y-\partial_t)u$ provides the required improvement in the coupled time-transport direction. Combining these estimates with the available control in the degree-three variables yields a full intrinsic improvement of flatness.

We prove sphere theorems for closed connected immersed hypersurfaces of constant scalar curvature in Euclidean space. Combining Ros's rigidity argument with the filling theorem of Huisken and Sinestrari, we show that every two-convex such hypersurface is a round sphere. As a consequence, Yau's conjecture holds for closed connected immersed hypersurfaces in \(\mathbb R^4\). Using Gromov's filling theorem, we also obtain higher-dimensional rigidity results under a principal-curvature index condition and an explicit scalar-curvature pinching condition.

The dimer model and random lattice permutations are two fundamental objects at the interface of probability, combinatorics, and mathematical physics. We study these models on finite periodic boxes in $\mathbb Z^d$ within a common framework. For the dimer model, edges connecting arbitrary vertices carry a weight which depends on their relative displacement and dimer configurations are weighted through their occupied edges. Superimposing two independent perfect matchings gives the double-dimer model, whose configurations are collections of disjoint loops. Permutations, instead, are weighted through the spatial displacement of their jumps. For broad classes of weights of finite or infinite range we prove long-range order and the occurrence of macroscopic loops. This extends nearest-neighbour results to arbitrary-range edges and jumps. In particular, long-range weights yield long-range order and macroscopic loops already in dimensions $d=1,2$. In dimension two, this behaviour is qualitatively different from that of the nearest-neighbour model. We complement these results with sharp absence criteria.

For graphs $H$ and $F$, let $\operatorname{ex}(n,H,F)$ denote the maximum number of copies of $H$ in an $F$-free graph of order $n$. Motivated by the Erdős-Sós theorem, Gerbner and Palmer and, independently, Zhao and Peng conjectured that for every tree $T$ of order $k$ and every $3\le r\le k-1,$ $$\operatorname{ex}(n,K_r,T)=a\binom{k-1}{r}+\binom{b}{r},$$ where $n=a(k-1)+b$ with $0\le b<k-1$. In this paper, we confirm the conjecture for $r\ge \left\lceil (2k-1)/3\right\rceil$ and characterize all extremal graphs.

We construct real Banach spaces $X$ and $Y$ and a bijection $U\colon X\to Y$ which preserves all pairwise distances on every closed ball of radius $1/4$ but is not a global isometry. The proof starts from an explicit injective map that preserves every distance at most $1/2$ and shortens one longer distance. We then enlarge the source and target through a transfinite recursion that inserts every missing target point while preserving both injectivity and this distance threshold.

Let $\mathsf{d}$ be the asymptotic density on the positive integers $\mathbb{N}^+$. We provide sufficient conditions on an infinite set $B\subseteq \mathbb{N}^+$ so that for each $\alpha \in [0,1]$ there exists a set $A\subseteq \mathbb{N}^+$ such that $\mathsf{d}(A)=0$ and $\mathsf{d}(A+B)=\alpha$. For instance, if $(b_n: n \in \mathbb{N}^+)$ is the increasing enumeration of $B$, it is sufficient that $\liminf_n b_{n+1}/b_n>1$. Moreover, we show that, if $|B\cap [1,n]|\ll \log(n)^{1-\varepsilon}$ for some $\varepsilon>0$, then for each $\alpha \in [0,1]$ there exists a set $A\subseteq \mathbb{N}^+$ such that $\mathsf{d}(A)=\mathsf{d}(A+B)=\alpha$. Conversely, we show that, for each $\varepsilon>0$, there exists an infinite set $B\subseteq \mathbb{N}^+$ such that $|B\cap [1,n]| \ll n^{2/3+\varepsilon}$ and $\mathsf{d}(A+B) \in \{0,1\}$ for every $A\subseteq \mathbb{N}^+$ such that $A+B$ admits asymptotic density.

Heavy-tailed $p$-value combination tests are attractive under unknown dependence because their null tails can be first-order robust even when the exact dependent null distribution is unavailable. That robustness does not resolve calibration: $\Pr\{T>q(\alpha)\}=\alpha+o(\alpha)$ neither quantifies the remaining size error nor determines its sign. We develop a second-order calibration theory for positive Half-Cauchy and reciprocal, or harmonic-mean, aggregation. After exact marginal standardization, extremal dependence is represented by an index-one exponent measure on the coordinate-face lattice. Its support separates axial, full-interior, proper-face, and mixed geometries. A Möbius decomposition shows how the noncompact weighted half-space probes every face, while quantitative face limits determine whether the correction is integrable, critically amplified, or nonintegrable. With common-factor inversion and a tail-to-calibration map, this yields machinery applicable beyond individual parametric copula families. The results include an integrable hidden-face transfer theorem, a sharp fixed-dimensional local-corner theorem with explicit higher-face control, a common-heavy-factor theorem, and size and critical-value expansions relative to independence. Gaussian, standard multivariate-$t$, positive Clayton, and max-linear pair-shock models exhibit distinct power, logarithmic, radial–angular, and proper-face mechanisms; the Gaussian weighted-half-space transfer is conditional on explicit face-boundary hypotheses. Thus dependence geometry determines the rate, coefficient, and direction of the calibration error left unresolved by first-order validity. Asymptotics use $t\to\infty$, equivalently vanishing significance levels.

This paper systematically investigates admissible configurations of the interaction between a rarefaction wave (RW) and a dispersive shock wave (DSW) by constructing a two-step piecewise constant initial condition in the Gerdjikov–Ivanov (GI) equation with non-convex hydrodynamics. Owing to the two-valued mapping associated with the Riemann invariants, the GI equation also supports contact DSWs (CDSWs), which are included in the interaction analysis. Using Whitham modulation theory, we elucidate the self-similar descriptions of the incoming waves, derive the boundary matching conditions, and obtain analytical expressions of the full evolution before, during, and after the interaction through generalized hodograph transformation. The results show that the interaction of a RW with either a DSW or a CDSW can exhibit two distinct scenarios: in one case, the two waves exchange spectral parameters after the interaction and fully separate, propagating independently with modified phases; in the other case, the wave structures merge upon interaction and subsequently evolve into either a soliton train or a small-amplitude harmonic wave for a classical DSW, and into an algebraic soliton train for a CDSW. The theoretical results are in good agreement with numerical simulations. This work can provide theoretical support for wave interaction phenomena in non-convex dispersive systems.

Spectrum governance requires rules to be translated into machine-executable decisions while preserving incumbent protection, regulatory authority, and an auditable record of why a decision was made. We present SPECTRA-GOV, a Tri-Layer Adaptive Governance Architecture (TLAGA) connecting international treaty coordination, national adaptive licensing, and real-time enforcement. The work builds on the original reference implementation preserved at v0.1.0-paper and adds a post-audit evaluation layer rather than replacing it. V2 introduces controlled spectral-contention scenarios, corrected geodesic-distance calculation, per-operator selective authorization, paired baseline counterfactuals, policy perturbation, regulatory-change analysis, and causal provenance. Across 10,000 scenarios in each of seven contention classes, incumbent protection was 100.00% in the no-contention control, 99.92-99.87% in weak-to-dynamic classes, 99.47% under strong overlap, and 90.00% in the adversarial close-proximity class. In a paired S3 baseline experiment, selective authorization achieved 100% incumbent protection and 89.43% access opportunity, while the population-wide dynamic baseline achieved 99.91% protection and 99.53% access. Identical results for the dynamic-SAS and SPECTRA-GOV selective variants mean selective admission alone is not claimed as an exclusive algorithmic novelty. The contribution is the integration of policy representation, computational enforcement, auditability, provenance. Enforcement timing was measured in-process, with mean latency increasing from 0.028 ms for one operator to 1.682 ms for 500 operators; these are software benchmarks, not field measurements. The results establish a reproducible computational governance prototype and identify remaining questions before operational or regulatory claims can be made.

For $0<a<1$, we study the linearization $L_a$ about a smooth odd self-similar collapse profile $(\Omega,c_l)$ of the generalized Constantin-Lax-Majda equation, where $c_l$ is the focusing exponent. On the origin-$H^2$ realization $X=\{\varphi\in L^2(0,\infty):\varphi(0)=0,\varphi''\in L^2(0,\infty)\}$, under explicit transport, regularity, weighted-derivative, and far-field hypotheses, the first three Edmunds-Evans essential spectra are exactly the union of two possibly coincident vertical lines: $\operatorname{Re}\lambda=F=-1+c_l/2$ at infinity and $\operatorname{Re}\lambda=O=-\tilde c/2$ at the origin, where $\tilde c=c_l+a(H\Omega)(0)$ and $H$ is the Hilbert transform. Off these lines, $L_a-\lambda$ is Fredholm, with index zero on the exterior components, $+1$ in the open strip when $F<O$, and $-1$ when $O<F$. The fourth and fifth Edmunds-Evans spectra and the Browder essential spectrum equal the closed strip. Every point of the open strip with index $+1$ is an eigenvalue. Every fixed point of the Huang-Qin-Wang-Wei positive-advection construction with $0<a<400/(848-9\pi^2)$, after normalization, satisfies the full hypothesis package. For all sufficiently small positive $a$, every such fixed point has $F<O$. Thus actual collapse profiles realize a nonempty Browder band. An additional kernel-nondegeneracy condition (ND) yields exact individual kernel and cokernel dimensions. We prove it outside an explicit vertical energy slab and, at sufficiently high frequency, on every closed vertical substrip of the open band separated from the far-field line. Failures in the open band are locally finite and can accumulate at high frequency only toward that line.

Let $K \subset \mathbb{R}^{m \times n}$ be a compact $C^1$-submanifold with boundary, $p \in (1,\infty)$ and $Q := (0,1)^n$. We prove that $K$ satisfies a rigidity estimate of the form $\|Du - (Du)_{Q}\|_{L^p} \leq C \|\mathrm{dist}_K(Du)\|_{L^p}$, $u \in W^{1,p}(Q,\mathbb{R}^m)$, if and only if $K$ satisfies sequential rigidity and for each $A \in K$, the tangent space to $K$ at $A$ satisfies exact rigidity. We further prove that this rigidity estimate is stable under small graphical perturbations of $K$. The key technical ingredient is proving that outside some small bad set where the maximal function of $\mathrm{dist}_K^p(Du)$ is large, the size of the superlevel sets of $\lvert Du - (Du)_Q\rvert$ decays exponentially. This is achieved by an adaptation of the John-Nirenberg inequality for $BMO$-functions.

This paper introduces and studies the Multi-Objective Enterprise Doubly-Green Supply Chain Network Design (MOE-DGSCND) problem, an extension of Multi-Objective Supply Chain Network Design. The problem studied here is a three-objective combinatorial optimization problem that simultaneously minimizes total cost f1, carbon dioxide emissions f2 and customer dissatisfaction f3 over binary decisions regarding which distribution centers to open and how to assign customer zones to open facilities. The problem generalizes the capacitated facility location problem (CFLP) and is known to be NP-hard (Garey and Johnson, 1979). This paper formally specifies the decision variables, objective functions and constraints and investigates three solution approaches using a single numerical instance: (A) an exact weighted-sum integer linear programming (ILP) formulation solved with Python combinatorially and via PuLP, (B) three Fully Polynomial-Time Approximation Schemes (FPTAS) providing (1 + $\epsilon$)-approximate solutions for $\epsilon$ := 0.15 and (C) two baseline heuristics for comparison. On an instance with eight customers and five candidate distribution centers, the exact solver certifies the optimal (sampled) supported solutions across eight weight combinations, while the FPTAS-based approaches complete within milliseconds, compared to centiseconds for the exact ILP on this tiny non-stochastic instance.

We study a genus-three hyperelliptic holomorphic null-curve family in $\mathbb{C}^4$, modelled on the algebraic data of the classical Schwarz P/D family, whose real parts define minimal immersions into $\mathbb{R}^4$. For the order-four automorphism $\omega\mapsto i\omega$ of the underlying Schwarz curve, we compute explicitly its action on the four holomorphic Weierstrass $1$-forms and derive the resulting identities for all real period vectors. Consequently, for every lattice invariant under the induced target rotation, torus-period closure can be checked on one representative from each orbit of a symmetry-stable homology generating set. We also identify precisely when the additional parameter produces a nondegenerate codimension-two deformation. The paper does not claim the construction of a new embedded periodic minimal surface in $\mathbb{R}^4$; rather, it provides an explicit symmetry reduction of the period problem associated with this family.

We study quantum potentials containing Dirac delta function (DDF) within its supersymmetric (isospectral) construction via a discontinuous superpotential, instead of being simply an added part. This construction comprises of two independently solvable sectors joined at the singular point by matching conditions imposed by the corresponding self-adjoint extension. Needing careful regularization, these boundary conditions impose independent algebraic constraints on each of the eigenstates, that restrict and truncate the spectrum. The number of surviving bound states, which are required to be parity-even, is thus fixed by the available system parameters rather than by the usual quantization rule. The specific cases of the Harmonic oscillator and the Rosen-Morse potential isospectrally infused with the DDF demonstrate such highly restrictive spectra, while the similar combination of the DDF with a Calogero-type singularity turns out to be largely incompatible as the regularization breaks down. Therefore, localized singularities are capable of controlling and engineering the discrete spectrum of quantum systems, with possible use in myriads of physical systems with very localized interactions.

We discuss Reeb spaces of non-proper smooth functions. (Non-)proper functions are maps between topological spaces the preimages of compact sets by which are compact (resp. may not be compact). Their Reeb spaces are the spaces of connected components of their level sets and with the natural quotient topologies. In proper cases, explicit theory are developing since the 1950s. This is closely related to Morse(-Bott) function theory and in such cases we have graphs naturally. Recently, related general topological studies with combinatorial ones are actively developing, mainly due to Gelbukh and Saeki, and they are at most $1$-dimensional in tame cases. We extend a theorem by Saeki in the 2020s: the Reeb space of a smooth function on a closed manifold is naturally a graph if and only if its critical value set is finite. The author has been a pioneer of the non-proper case and previously investigated examples.

Under individual smoothness, known algorithms for nonconvex finite-sum optimization use $O(n+\sqrt n\Delta L_{\max}/\varepsilon^2)$ incremental first-order oracle (IFO) calls, while existing lower bounds miss the factor $\sqrt n$ in the second term. We prove the matching lower bound $\Omega(n+\sqrt n\Delta L_{\max}/\varepsilon^2)$ for randomized IFO algorithms, allowing component indices and query points to depend on the complete preceding history. Thus PAGE and SPIDER are minimax optimal up to universal constants under individual and mean-squared smoothness. Under the global Polyak-Lojasiewicz (PL) condition, prior lower bounds did not capture the correct dependence on $n$ under individual smoothness, and the regime of small $\kappa$ remained open for randomized IFO algorithms. We prove an $\Omega(n+\kappa_{\max}\sqrt n\log(\Delta/\varepsilon))$ lower bound for $\kappa_{\max}\ge\sqrt n$, and an $\Omega(n+n\log(\Delta/\varepsilon)/(1+\log(\sqrt n/\kappa_{\max})))$ lower bound for $3\le\kappa_{\max}\le\sqrt n$. Restarted PAGE attains matching upper bounds under mean-squared smoothness in terms of $\kappa_{\rm ms}=L_{\rm ms}/\mu$. Our lower bounds use dense weak hiding. Each hidden direction is spread across all components, making individual queries weakly informative while preserving the direction in the average. A sufficiently large dimension makes premature activation of future stages unlikely within the query budget. Balancing the cost of revealing a stage against the number of stages permitted by individual smoothness yields the missing $\sqrt n$ factor.

Let $A$ be an $n$-element set of positive real numbers, let $FS(A)$ be its set of subset sums, and put $T_n=\binom{n+1}{2}$. For every fixed integer $C\geq-1$ and all sufficiently large $n$, we classify the sets satisfying $$ |FS(A)|\leq T_n+n+C+1. $$ Each such set is commensurable. Its unique primitive integer normalisation $B$ either satisfies $\sum B\leq T_n+n+C$ or belongs to an explicit exceptional family specified by a missing element $m\in\{1,2\}$ and an integer partition of $C+m$ or $C+m+1$. If $P$ denotes the partition function, the exceptional family has exactly $$ P(C+1)+2P(C+2)+P(C+3) $$ primitive dilation classes. We also prove a local inverse theorem for bounded increment excess. If, for sufficiently large $i$, adjoining the largest element to the preceding $i-1$ elements creates only $i+e$ new subset sums, where $e$ is bounded, then the $i$-element set is a dilation of $[1,i+e]_{\mathbb{Z}}$ with exactly $e$ elements deleted. Conversely, every such deletion pattern has increment excess $e$. The proof combines a stabiliser argument in $\mathbb{R}/x\mathbb{Z}$, Kneser's theorem, a quadratic subset-sum bound, and endpoint propagation. These arguments also give effective commensurability and a finite-state encoding. Together with earlier results for $C\leq-2$, this completes the eventual fixed-defect classification for every integer $C$.

Multistable mechanical systems can store and release elastic energy through snap-through instabilities, but controlling transition pathways between stable states remains challenging when multiple routes are accessible. Here, we introduce a two-mass von Mises truss as a general model for studying pathway selection governed by coupled saddle-node bifurcations. The system consists of two coupled snap-through units with geometric imperfections, giving rise to four stable configurations: a fully inverted state, a fully natural state, and two intermediate mixed states. We show that the coupling stiffness reorganizes the quasi-static bifurcation structure and selects among three transition pathways under release: sequential snapping through one mixed state, direct cooperative snapping, or sequential snapping through the other mixed state. Using pseudo-arclength continuation, we track the relevant saddle-node bifurcations and identify the parameter regimes associated with each quasi-static pathway. We then demonstrate that dynamic bifurcation delay provides an additional rate-dependent mechanism for pathway selection. Even when the quasi-static bifurcation structure favours a unique sequential pathway, finite-rate loading delays snap-through beyond the corresponding static saddle-node points and can reorder the snapping sequence of the two units. A local reduction of the coupled dynamics near each saddle-node yields normal forms with coupling-dependent critical points and coefficients. The resulting theory identifies distinct rate-dependent delay laws in the inertia-dominated and overdamped regimes and predicts the critical rate at which the snapping order reverses. These results establish a general mechanics framework for tuning transition pathways in multistable systems through elastic coupling and loading-rate control.