Papers

In this paper, we study de Rham realizations for finite orientifold quotients. On the orientation preserving inertia of an almost complex global quotient we construct an anti-linear involution of the Chen-Hu algebra whose fixed subalgebra is a real form of the Chen-Ruan algebra. For genuine orientation reversing fixed sectors we prove that no coordinate invariant single sector fractional age exists. When a Calabi-Yau volume line is preserved, the relative phases define a canonical class $[\mu_{\mathbb R}]\in H^2(\widehat G;\mathbb Z_\epsilon)$. We identify this class with the Bockstein obstruction to a coherent real phase lift and prove that its vanishing is equivalent, after rephasing the volume form, to a phase trivial orientifold action. We then formulate a general Euler-Gysin realization theorem using relative Fredholm bundles and determinant-line sewing. For phase trivial Calabi-Yau surfaces we verify the admissibility conditions directly and obtain a canonically defined associative graded cohomology algebra containing the Chen-Hu even sector and the odd fixed surfaces. For an Eisenstein abelian surface with an order six dihedral action we compute the centralizer invariant algebra explicitly; it is a $16$ dimensional Frobenius algebra whose odd sector multiplication is noncommutative in the shifted grading. We also explain the relation with earlier unoriented orbifold cohomology constructions.

We develop an interdisciplinary framework for evaluating the epistemic robustness of social-scientific inquiry through Hadamard's criteria for well-posed problems: existence, uniqueness, and stability. Reinterpreting these criteria not as demands for deterministic certainty but as methodological guardrails, we show how they diagnose recurrent failures of identification and inference across quantitative and qualitative paradigms. In quantitative research (e.g., econometric modeling), instability manifests when substantive conclusions depend sensitively on model specifications or data filtering. In interpretivist qualitative research, non-uniqueness and non-falsifiability arise when theoretical frameworks are elastic enough to accommodate contradictory observations without pre-specified rejection criteria. We frame these failures as inverse problems where the mapping from empirical data to substantive claims fails to satisfy existence, uniqueness, or stability. Finally, we propose cross-paradigmatic safeguards: ex-ante falsification criteria, empirical boundary conditions, multiverse sensitivity auditing, and cross-observer validation, to ensure social-scientific claims remain appropriately constrained, falsifiable, and robust to perturbations in evidence and interpretation.

Motivations and derivations for Heron's triangle area formula abound. But how would one discover the formula? We recount the challenge of designing a laboratory experiment to discover Heron's formula, and of doing so in the morning, in time for a noon class—a snapshot of collaboration with AI in fall 2025. We then reflect on extensions and generalizations.

We conjecture an explicit evaluation of the pure fifth Bessel moment $\int_0^\infty K_0(t)^5\,dt$ as a quadratic expression in the critical value $L(f,2)$ of the weight-three, level-60 newform $f$ (LMFDB orbit 60.3.b.a) identified in the twisted fifth-moment modularity theorem of Lim, Tu and Yu, with coefficients in $\mathbb{Q}(\sqrt{5})$ and the square taken before the real and imaginary parts. Directed interval computations, using no stored Bessel or $L$-values, bound the absolute discrepancy by $10^{-358}$. We prove three exact modular identities for $f$: the Petersson-norm formula $\langle f,f\rangle_{60} = \frac{3(5-\sqrt{5})}{2\pi^4}|L(f,2)|^2$, the coefficient-conjugation relation $L(f^{\sigma},2) = \kappa L(f,2)$ with explicit $\kappa \in \mathbb{Q}(\sqrt{5},i)$, and the twisted symmetric-square evaluation $L(\chi_{-4}\mathrm{Sym}^2 f,2) = \sqrt{15}\,\pi^2 \langle f,f\rangle_{60}$, together with $L(\chi_{-4}\mathrm{Sym}^2 f,3) = \pi^4\langle f,f\rangle_{60}/8$, in the full Euler-factor normalization of Lim, Tu and Yu. The last identity shows that the companion norm conjecture $D_{5,\mathrm{odd}} = \frac{3\sqrt{15}(5-\sqrt{5})}{2}|L(f,2)|^2$ is equivalent to the symmetric-square conjecture $D_{5,\mathrm{odd}} = \pi^2 L(\chi_{-4}\mathrm{Sym}^2 f,2)$ of Lim, Tu and Yu, while the exact relation $D_{5,\mathrm{even}} = \pi^2 D_{5,\mathrm{odd}}/(2\sqrt{15})$ follows from Chuang's period formulas. Every Bessel-to-modular equality, including the individual-period formula, remains conjectural. Complete proofs, exact rational certificates and verification programs are included as ancillary files.

In this article, we prove the irrationality of at least one of the three numbers $\zeta(5), \zeta(7)$ and $\zeta(9)$ is irrational. We find linear forms in $1,\zeta(5),\zeta(7)$ and $\zeta(9)$ and prove their arithmetic by Kranttenthaler and Rivoal. We also refine these estimates by using integral quantity $\Phi$ involving specific product of prime powers. We use the results of Kranttenthaler, Rivoal and Zudilin to prove that at least one of the three numbers $\zeta(5), \zeta(7)$ and $\zeta(9)$ is irrational.

Let $\Omega$ be the curvilinear square bounded by the four circular arcs $|\zeta\mp1|=\sqrt{2}$, $|\zeta\mp i|=\sqrt{2}$, and let $W=\sqrt{1+14\zeta^{4}+\zeta^{8}}$. Put \[ \theta(\zeta)=\arcsin\frac{2(1+i)\zeta} {\sqrt{1+4i\zeta^{2}-\zeta^{4}}}, \] \[ f=\tfrac14\bigl(-iF[\theta,\tfrac14]+F[\theta,\tfrac34]\bigr),\quad \quad g=\tfrac14\bigl(\ \ iF[\theta,\tfrac14]+F[\theta,\tfrac34]\bigr), \] \[ \mathfrak h=(2-\sqrt3)\, F\bigl[\arcsin\bigl(i(2+\sqrt3)\zeta^{2}\bigr),(2-\sqrt3)^{4}\bigr]. \] These are, verbatim, the three functions of our companion paper $\mathrm{[D]}$ on the diamond surface $\mathrm{D}$: the two papers start from the same point, and differ only in which real parts are taken. With $X=(\kappa\,\operatorname{Re}f,\ \kappa\,\operatorname{Re}g,\ \tfrac12+\kappa\,\operatorname{Im}\mathfrak{h})$ and $\kappa=3/(2K[1/9])$, the map $X$ parametrizes a fundamental patch of Schwarz's primitive surface $\mathrm{P}$, the conjugate of $\mathrm{D}$. Part I builds the algebraic apparatus $\ldots$ the normalizing constant is identified exactly: $\varpi:=\kappa^{-1}=K[-3]=\tfrac12 K[3/4]=\tfrac23 K[1/9]$. Part II carries out the elimination and completes the proof of \[ (\star)\qquad \operatorname{sn}\bigl(\varpi(x{+}y),-3\bigr)\,\operatorname{sn}\bigl(\varpi(x{-}y),-3\bigr) =\tfrac13\operatorname{sn}\bigl(3\varpi(z-\tfrac12),\tfrac19\bigr). \] Part III shows that $(\star)$ is, after all, additively separable $\ldots$ for an overview of this paper and $\mathrm{[D]}$, please see arXiv:2609.14206.

The representation number of a graph is the smallest integer $k$ such that the graph can be represented by a word in which each vertex appears exactly $k$ times, and two distinct vertices $x$ and $y$ alternate in the word if and only if they are adjacent in the graph. We extend known results on the representation number for various graph classes to chessboard graphs - namely, king, queen, rook, bishop, and knight graphs. We provide a complete classification for queen graphs and partial classifications or observations for the other classes. As a consequence of our study, we obtain a characterization of all chessboard graphs that are circle graphs. Our work also leads to several interesting open problems.

Coupling-matrix synthesis of cross-coupled resonator filters is difficult when the desired topology is noncanonical. Automatic differentiation removes the need for problem-specific sensitivity expressions, but does not by itself overcome the nonconvex optimization landscape. This work develops a staged differentiable synthesis framework and identifies the importance of its initial objective through a controlled comparison that varies only the objective of the first optimization phase. For tenth- and sixteenth-order benchmarks, directly matching a frequency-sampled response under the prescribed topology reaches the target from $0$ of $24$ and $0$ of $12$ random starts, respectively. Replacing the first-stage objective with characteristic-polynomial matching increases the success rate to $18$ of $24$ and $12$ of $12$ (exact paired $p=7.6\times10^{-6}$ and $4.9\times10^{-4}$). The observed difference is not explained by local conditioning and is consistent with saturation of the bounded response residual far from the solution. A differentiable front end converts symmetric multiband specifications into target polynomials and, when applied end to end to a fabricated dual-band filter specification, finds a realization with one fewer cross-coupling than the published design. Finally, differentiating through Monte Carlo tolerance simulations enables yield-driven design centering, increasing the model-predicted yield from $50.9\%$ to $67.0\pm0.5\%$ under an additive coupling-error model, an improvement of $16.1$ percentage points.

In this paper, the unified generalized $q$-Stirling matrices are defined in terms of the $q$-analogues of the unified generalized Stirling numbers. Several matrix relations involving these matrices are derived, and matrix decompositions of the unified generalized $q$-Stirling matrices are established.

Let $5\leq p_1<\cdots<p_r$ be primes, put $P=p_1\cdots p_r$, and let $g_P(2n)$ count the unordered decompositions $2n=h+k$ for which both summands are coprime to $6P$. We derive an explicit finite inclusion–exclusion formula for $g_P(2n)$ by parametrizing the admissible residue classes and combining the local restrictions through the Chinese remainder theorem. The associated complete-modulus correlation is a specialization of Nagell's totient function and determines both the finite local density and the affine increment $g_P(2(n+6P))-g_P(2n)=\kappa_P(n)$. A parameter-density decomposition gives a residue-sensitive positivity criterion and the uniform sufficient condition $n\geq 3(3^r-1)\prod_{i=1}^{r}\frac{p_i}{p_i-2}+2\Longrightarrow g_P(2n)>0$. Maximizing this bound over squarefree moduli whose prime factors do not exceed a prime $p$ gives an explicit threshold $\mathcal N(p)$ satisfying $\log\mathcal N(p)\sim(\log3)p/\log p$. We also describe the complete prime cutoff at $\sqrt{2n}$ and its exact additive-prime interpretation. We also evaluate $g_P(2tP)$ explicitly using the real character of period $3$. The positivity thresholds are elementary and fully explicit; lower-bound sieves provide smaller asymptotic scales, and the present thresholds do not establish positivity at the complete cutoff for all $n\geq13$.

The clique partition number of a graph is the minimum number of complete subgraphs whose edge sets partition its edge set. We study graphs in which, in every induced subgraph and outside every prescribed clique, some vertex has a neighbourhood that becomes a clique after deleting at most $s$ vertices. The case $s=0$ is exactly the class of chordal graphs. For each fixed $s$, we prove that the maximum clique partition number at all sufficiently large orders $n$ is $\lfloor(n+s)(n+s+1)/6\rfloor-\binom{s+1}{2}$, and determine all equality graphs. The same expression is an upper bound up to an additive constant depending only on $s$ at every order. In particular, every chordal graph has clique partition number at most $n^2/6+n/6+O(1)$, answering a question of Erdős, Ordman and Zalcstein. We also prove structural stability for sublinear defect and a sharp finite-order theorem for integer signed clique functionals. The proof combines signed fractional localization with an edge-disjoint triangle construction; only a qualitative fractional-packing approximation is required.

For a ferromagnetic Ising model on a graph of maximum degree $\Delta\ge3$, we prove a bound of order $\sqrt n$ on every row of the influence matrix at the tree uniqueness threshold. The estimate is uniform in the degree, the external fields, and all pinnings. More generally, if the couplings are bounded by $\beta$ and $\varepsilon=((\Delta-1)\tanh\beta-1)_+$, the bound is $C(\sqrt n+n\varepsilon)$. The proof combines a pointwise cavity bound with a positive-series magnetization tilt and the field comparison theorem of Ding, Song and Sun. The critical estimate removes the logarithm in recent general graphical bounds for the ferromagnetic case. As a consequence, zero-field single-site Glauber dynamics mixes in polynomial time throughout the supercritical window $\varepsilon=O(\sqrt{\log n/n})$, with the polynomial degree depending on the window size.

For \(n\ge3\) and a graph \(H\) on at most \(n\) vertices, let \(q(H)\) be the expectation threshold for its containment in \(G(n,p)\). We prove that there are absolute constants \(a,L>0\) such that, for every \(C>0\), every graph of degeneracy at most \(C\log n/\log\log n\) has a deterministic edge decomposition into at most \(\lceil a(C+1)\rceil\) pieces, each with ordinary containment threshold at most \(Lq(H)\). This removes the maximum-degree hypothesis from a theorem of Ascoli, He, Park, and Talagrand in their approach to Talagrand's discrete convexity problem. With constants depending on the fixed parameters, a structural extension allows the addition of \(O(\log n)\) vertices with arbitrary incident edges. The uniform main bound also gives an \(O(1+\log\log n)\)-piece decomposition for every target graph, with an absolute threshold multiplier. The main ingredient converts Li's two-set coupling for spread measures into a partition of a prescribed neighborhood list, fixed before the random host is sampled. It allows arbitrary overlaps and repetitions and controls all Hall matching conditions simultaneously by bounding target demand and host supply through common two-sided approximations under the biased host measure.

For a finite simple graph G, let alpha(G) denote its independence number and let l_avg(G) = (1 / |V(G)|) sum_{v in V(G)} alpha(G[N_G(v)]) be the average independence number of its open neighbourhoods. Written on the Wall II Conjecture 194 asserts that every simple connected graph on n > 1 vertices satisfying alpha(G) <= 1 + l_avg(G) has a Hamiltonian path. We give a four-parameter family of counterexamples. Its principal two-parameter subfamily satisfies the proposed inequality with equality: for every pair of integers s >= 1 and t >= 3 it has (s + 1)t^2 vertices, independence number t + 1, l_avg(G) = t, and minimum degree s, but has no Hamiltonian path. This entire infinite subfamily is machine-checked in Lean 4: one universally quantified theorem certifies its order, connectivity, independence number, average neighbourhood independence, minimum degree, conjecture hypothesis, and failure of traceability. Thus no fixed lower bound on the minimum degree repairs the conjecture. The case (s,t) = (1,3) has 18 vertices, but the formal certificate is parametric rather than a verification of that one graph alone.

We study umbilic slopes and the global classification of cubic Weingarten surfaces. For a smooth surface in Euclidean three-space, a nonconstant principal curvature germ with a unique secant tangent has slope zero, minus one, infinity, an odd integer at least three, or its reciprocal. At a nonconstant umbilic germ, a smooth regular relation between mean and Gaussian curvature supplies such a tangent after a continuous labeling of the principal curvatures; the ordered curvature image has at most two limiting directions, both in the same set. We also classify compact immersed surfaces without boundary of class $C^4$ satisfying $\kappa_2=c\kappa_1^3$ in some order at every point, for a fixed real $c$: each connected component is embedded as an ellipsoid of revolution. The arguments share a classification of homogeneous Hessian identities at an umbilic. Scalar unique continuation treats infinite-order contact in the smooth slope problem, whereas an exact fourth-order growth estimate and angular energy estimates yield the cubic classification.

Let $z_{n,j}(\lambda)$ denote the positive zeros, in decreasing order, of the ultraspherical polynomial $C_n^\lambda$, $\lambda>-1/2$, with the reduced limiting interpretation at $\lambda=0$ specified below. Our principal result settles three higher-monotonicity questions of Gautschi: two as printed and the natural open-interval form of the third, whose printed endpoint $\lambda=0$ is singular. For every $n\geq3$, $$ \sqrt{\lambda+1}\,z_{n,j}(\lambda) $$ becomes a complete Bernstein function after translation of its parameter interval to $(0,\infty)$. For every $n\geq2$, $$ \sqrt{\lambda}\,z_{n,j}(\lambda) $$ is a complete Bernstein function on $(0,\infty)$. For every $n\geq4$, the largest-zero trajectory $$ \sqrt{\lambda+\frac{2n^2+1}{4n+2}}\,z_{n,1}(\lambda) $$ has the same property, whilst for every other positive zero the derivative of this scaling fails to be completely monotone. Separately, an exact calculation in degree $4$ provides a counterexample to a fourth conjecture of Gautschi, concerning the linear scaling $\lambda z_{n,j}(\lambda)$.

Let $1 < \rho < n$ and let Omega be in $L^\rho(S^{(n-1)})$ with vanishing mean. We prove that the maximal truncation $T^*_\Omega$ of the rough singular integral $T_\Omega$ is pointwise dominated by finitely many sparse potentials of the form: $\sum_{Q \in S} l(Q) * ( (1/|Q|) * \int_Q |\nabla f|^p )^{1/p}$, where $1/\rho~ = 1/\rho' + 1/n$ and $\rho~ \leq p < n$. This estimate is uniform in the truncation parameter and extends the subcritical bound of Hoang, Moen, and Perez for $T_\Omega$ to $T^*_\Omega$. Since the argument does not require the boundedness of $T^*_\Omega$ on the target space, it yields two-weight Sobolev inequalities: $\parallel T^*_\Omega f \parallel L^q(u) \leq C * \parallel \nabla f \parallel L^p(v)$, under joint two-weight conditions, while the target weight u itself is only required to belong to $A_\infty$. Additionally, we prove a Hedberg-type estimate involving a Morrey norm of the gradient and apply it to weighted grand Lebesgue spaces. Further consequences are obtained in weighted Lebesgue, Orlicz, and variable Lebesgue spaces.

We consider interaction functionals of the form \begin{equation*} \nu\mapsto {\mathscr E}(\nu)=\int\_{{\mathbb R}^N}\int\_{{\mathbb R}^N} W(x-y)\intd\nu(y)\intd\nu(x)+\int\_{{\mathbb R}^N} V(x)\intd\nu(x)\text{,} \end{equation*} involving an anisotropic kernel $W$ and a general confinement potential $V$. Under standard assumptions on $W$ and its Fourier transform, we show that the minimizer of ${\mathscr E}$ has $L^2$ density. Our argument relies on a careful analysis of nonlocal variational inequalities, which may be of independent interest.

For integers $n,r,t$ with $2\le t\le r-1$, let $f(n,r,t+1)$ denote the largest possible minimum degree of a balanced $r$-partite graph with parts of size $n$ and containing no copy of $K_{t+1}$. Lo, Treglown and Zhao identified $f(n,7,4)$ as the only remaining case in their treatment of the $K_4$-free family. I determine this function for every $n\ge1$. First, the corresponding three-colourable extremum $\delta(n,7,3)$ is reduced to a $7\times3$ integer matrix problem and determined exactly. Second, a structural argument shows that every balanced $7$-partite $K_4$-free graph $G$ with $\delta(G)>\frac{132}{31}n$ is three-colourable. Consequently, $f(n,7,4)=\lfloor30n/7\rfloor$ except when $n\equiv4\pmod7$ and $n\ge11$, where it is one less.

Exploiting the equality restrictions that nested Markov models encode requires their tangent-space geometry. For strictly positive finite-state models on arbitrary acyclic directed mixed graphs (ADMGs), we differentiate the recursive-head chart and prove that the range of its score map is the full tangent space. Intrinsic-set coordinate blocks form an algebraic direct sum, blocks of distinct districts are orthogonal, and the resulting Gram projection needs neither mb-shieldedness nor a district order. Exact counterexamples show that observational centering and kernel normalization alone do not certify tangency. For the node, complete-source edge, and compatible path-specific intervention targets considered here, boundary substitution yields a normalized configured active law and an order-free canonical-gradient formula, extended to finite mixtures by independent source redraws. A coherent one-step estimator with an exact remainder identity and a model-valid chart-flow targeted maximum likelihood estimator are efficient under stated local conditions. On a narrower source-isolated fixed-node subclass, a sequential estimator has an exact transition-factorized drift and up to $2^m$ nuisance-correctness regimes over $m$ active-district transitions; its all-correct influence function projects onto the canonical gradient, and an exact rational law exhibits a strict variance gap. These results separate order-free efficiency on arbitrary finite-state ADMGs from multiple robustness of a narrower construction.

The kinetic Motsch-Tadmor (in short, KMT) model is a kinetic flocking model with a normalized communication weight. In this paper, we study the emergent dynamics of the phase-spatially extended KMT model. We first establish a global well-posedness theory in the fully noncompact spatial-velocity setting. To this end, we introduce a direct Lagrangian formulation in which the unbounded part of the initial velocity distribution is separated from an interaction-generated bounded remainder. This decomposition allows us to construct global Lagrangian weak solutions without truncating the velocity distribution and to propagate finite phase-space moments. We then investigate the long-time collective behavior of the resulting solutions. When the initial velocity support is compact while the spatial support is allowed to be noncompact, a time-varying effective-region argument yields a uniform contraction mechanism for the normalized interaction and leads to exponential weak support flocking. When both the spatial and velocity supports are noncompact, support-level flocking is in general impossible. Nevertheless, the same effective-region mechanism, combined with the Lagrangian decomposition and a bootstrap argument for the interaction-generated remainder, yields exponential weak moment flocking. In particular, pairwise spatial moments remain uniformly controlled while velocity fluctuations converge exponentially to zero. These results provide a unified framework for the well-posedness and flocking dynamics of the KMT model beyond the compact-support regime.

The demand matching problem generalizes both the knapsack problem and the $b$-matching problem. In this problem, each edge of a graph has a demand and a weight, and each vertex has a capacity. The goal is to find a maximum weight subset of edges such that, at each vertex, the total demand of the incident selected edges does not exceed the vertex capacity. Parekh [IPCO 2011] proved that, if each edge is individually feasible, the natural LP relaxation for demand matching has integrality gap at most $3$, yielding a $3$-approximation algorithm. This bound is tight for the natural LP relaxation, matching the lower bound of Shepherd and Vetta [Math. Oper. Res. 2007]. We present a randomized $(3/2 + \sqrt{2} + \varepsilon) \approx (2.914 + \varepsilon)$-approximation algorithm for the demand matching problem for every $\varepsilon > 0$, giving the first approximation ratio strictly better than $3$. For bipartite graphs, we obtain a randomized $(2 + \varepsilon)$-approximation algorithm for every $\varepsilon > 0$. Both algorithms run in time polynomial in $1/\varepsilon$ and the input length. Our algorithms use a strengthened LP relaxation based on intersecting the integral knapsack polytopes associated with the vertices, together with a multiple-choice generalization. As a key ingredient, we prove the existence of a $(q, 1/(1+q))$-balanced contention resolution scheme for the integral knapsack polytope for every $q \in [0, 1]$, which may be of independent interest. The balance guarantee $1/(1+q)$ is tight in the worst case over all knapsack instances.

We study the radical and screen distributions of a lightlike hypersurface of an indefinite statistical manifold with respect to the induced connections arising from the ambient dual affine connections. Necessary and sufficient conditions are obtained for each distribution to be parallel with respect to both induced connections. These conditions are expressed through the screen shape operators, the local screen fundamental forms, the mean induced connection, and the tangential difference tensor. When both distributions are parallel, we characterize the screen projection and obtain a block decomposition of the tangential difference tensor, whose mixed radical-screen part vanishes. We also obtain equivalent conditions for the induced connections to be metric connections. In the metric case, the induced screen connections coincide with the Levi–Civita connection on each integral manifold of the screen distribution, and the curvature tensors of the three induced connections agree on screen directions. We further determine the action of these curvature tensors on the radical direction. An explicit example shows that both distributions may be parallel although neither induced connection is a metric connection and their curvature tensors need not vanish.

This paper proposes a deterministic bilevel optimization framework for the coordinated operation of Virtual Power Plants (VPPs) embedded in an active distribution network. The Distribution System Operator (DSO) acts as the upper-level leader, minimizing a weighted combination of expenditure, active losses and voltage deviation subject to nonlinear AC power flow constraints, while each VPP operates as a lower-level follower that maximizes its profit under the uniform price signal issued by the DSO. Unlike most existing formulations, which linearize the lower-level subproblem to obtain a Mixed-Integer Linear Program (MILP), the proposed model retains the full AC Optimal Power Flow (AC-OPF) equations, producing a bilevel Mixed-Integer Nonlinear Program (MINLP). The lower-level problem is replaced by its Karush-Kuhn-Tucker (KKT) optimality conditions and the Strong Duality Theorem, yielding a single-level Mathematical Program with Equilibrium Constraints (MPEC). Complementarity conditions are then linearized via the Fortuny-Amat big-M transformation. The framework is validated on the IEEE 33-bus feeder over a 24-hour horizon, with four distributed resources aggregated into a single VPP. Compared with individual dispatch against a regulated time-of-use tariff, aggregation reduces active losses by 10.5 %, the accumulated voltage deviation by 18.3 %, and the bus-hours below 0.95 p.u. from 132 to 29. These gains cost 0.31 % in social cost and 0.26 % in DSO expenditure, while the rent of the aggregator is preserved.

For a collection of states indexed by $[n]={1,\ldots,n}$, let $F(H)$ denote the optimal unnormalized minimum discrimination error for each nonempty subfamily $H\subseteq[n]$. We call $F$ the discrimination profile and characterize exactly which profiles can arise in a general probabilistic theory (GPT). For fixed $n$, the GPT region $\mathfrak{G}_n$ is a bounded rational polytope, giving a hierarchy $\mathfrak{C}_n\subseteq\mathfrak{Q}_n\subseteq\mathfrak{G}_n$ of classical, quantum, and GPT profiles, analogous to the local–quantum–no-signalling hierarchy in Bell theory. For $n=3$ we determine the classical and GPT polytopes explicitly. Within $\mathfrak{G}_3$, the classical region is characterized by $\beta:=F(123)-F(12)-F(13)-F(23)\le0$, while the GPT maximum is $1$. For qubit triples, the maximal value is attained by the trine, $\beta_{\mathrm{tr}}=3\sqrt3/2-2$, and we prove the dimension-independent quantum bound $\beta_{\mathrm Q}\le2/3$, with a slight further improvement. Hence $0=\beta_{\mathrm C}<3\sqrt3/2-2\le\beta_{\mathrm Q}\le2/3<1=\beta_{\mathrm{GPT}}$. We conjecture $\beta_{\mathrm Q}=\beta_{\mathrm{tr}}$ and prove this for arbitrary pure-state triples, together with further supporting evidence. For arbitrary $n$, every classical facet admitting a GPT violation already admits a quantum violation in dimension at most $3$. A four-state qubit example shows that quantum success profiles need not be submodular, unlike for $n=3$. Finally, we exhibit a four-dimensional quantum triple whose discrete profile is classical for all subfamilies but becomes nonclassical when unequal priors are allowed.

For a constraint satisfaction problem defined by a relation $R$, its non-redundancy $\text{NRD}(R,n)$ is the size of largest instance (as a function of the number $n$ of variables) for which no constraint is implied by the rest. Its chain length $\text{CL}(R,n)$ is the largest such instance where the constraints can be ordered so that no constraint is implied by the preceding ones. Clearly $\text{CL}(R,n) \ge \text{NRD}(R,n)$ but so far no asymptotic separation was known between these quantities. We exhibit an explicit arity $4$ relation for which $\text{CL}(R,n) \ge \omega(\text{NRD}(R,n))$.

We find Sasaki-Einstein metrics on rational homology $(4n-1)$-spheres for $n>1$ built from cyclic polynomials of index 1 cutting out rational varieties. The Einstein metrics found here are inequivalent to the ones found by Boyer and Galicki in arXiv:math/0311355. Our findings are consequence of an improvement, for hypersurfaces defined by cycle polynomials, on the estimate given by Johnson and Kollár to determine Kähler-Einstein orbifold metrics. We also construct weighted hypersurfaces that contain the rational varieties described above as codimension two subvarieties and, due to the refined estimate for cyclic polynomials, we find conditions on the weights and degrees of these hypersurfaces so their corresponding smooth links admit Sasaki-Einstein metrics. Finally we study the effect of the Berglund-Hübsch transpose rule on the topology and on the existence of Sasaki-Einstein metrics on the links studied and generalize all the results given in arXiv:2311.15998 for rational homology 7-spheres to rational homology $(4n-1)$-spheres, that is, we show invariance of these two features under the transpose rule.

We extend the discrete complex complement quotient (DCCQ) framework from binary Bernoulli counts to multinomial count compositions. For m+1 categories, m is the number of independent probability degrees of freedom. Integer count vectors modulo common scaling determine rational points of the m-dimensional probability simplex. Building on standard simplex and log-ratio coordinate geometry, for m >= 2 we define the full multinomial DCCQ coordinate map and show that it is a real-analytic diffeomorphism The previously established binary baseline m=1 gives a critical-line coordinate, while the ternary case m=2 gives the full open critical strip; higher multinomial models retain m-2 additional real contrasts. We also give a one-versus-rest specialization, an exact integer-lattice realization of the ternary coordinate, and a hyperbolic representation of its log-ratio. No zero-location theorem or proof of the Riemann Hypothesis is claimed.

We study toric vector bundles on complete toric varieties whose total equivariant Chern classes are trivial. Our approach is tropical, using the notion of tropical toric vector bundles as piecewise linear maps introduced by Kaveh and Manon. We prove that any toric vector bundle of rank $r$ with trivial Chern class and affinely independent equivariant Chern roots is equivariantly isomorphic to a toric vector bundle pulled back from one of a finite set of varieties with dimension at most $r-1$ after twisting by a character. This extends a theorem of Payne about toric vector bundles of rank $r\leq 3$ with trivial Chern class. As an application, we construct examples of complete toric varieties of dimension $n$ that admit no nontrivial toric vector bundles of rank $r\leq n+1$ with the aforementioned properties. We also introduce combinatorial gadgets we call flag decorations of permutohedra, whose convexity properties are key for our results.

Eternal Vertex Cover (EVC) is a turn-based attacker-defender game on an undirected graph $G$. To begin with, the defender places $k$ guards on vertices of $G$. The attacker, on their turn, can choose an edge $e$ not already occupied at both endpoints to "attack". The edge $e$ is defended if a guard moves along the edge $e$. The defender, on their turn, can move any subset of guards. A guard can only move to a neighboring vertex. The minimum number of guards needed to indefinitely defend against any sequence of attacks is called the eternal vertex cover number, generalizing the classic vertex cover number. Determining this number is NP-hard in general, motivating the study of parameterized and approximation algorithms. The problem is known to be FPT when parameterized by the cover number, but structural parameters remain relatively unexplored in the literature. In this work, we explore structural parameterizations for EVC. We show that EVC is FPT parameterized by the cluster vertex deletion number, which generalizes the previously studied parameterization by vertex cover number. We next study the problem parameterized by vertex integrity, which is the smallest number of vertices we need to delete from $G$ so that the resulting graph is a disjoint union of constant-sized components. We first show that Eternal Vertex Cover is XP parameterized by vertex integrity. Then, we develop a polynomial-time approximation algorithm, which computes an additive $6k+1$ ($g(k)$) approximation, where $k$ is equal to the cluster vertex deletion number (vertex integrity). Finally, we show a FPT algorithm for when the deletion set produces "nice" connected components, which are components that are bounded in size and satisfy a technical condition.