Papers

Let $(X,\rho,\mu)$ be a space of homogeneous type satisfying $\mu(X) =\infty$, the doubling property and the reverse doubling condition. Let $L$ be a nonnegative self-adjoint operator on $L^2(X)$ whose heat kernel has a Gaussian upper bound. First, we study the preduals of weighted Bourgain-Morrey spaces on $X$ and their properties, such as lattice property and completeness. Then we introduce the weighted homogeneous Besov-Triebel-Lizorkin block spaces. It is shown that they are the preduals of weighted homogeneous Bourgain-Morrey Besov-Triebel-Lizorkin spaces associated with the operator $L$. Consequently, these block spaces are independent of the choice of partitions of unity.

Branch model predictive control optimizes multiple future trajectories coupled through shared decisions, with computational demands increasing as the number of scenarios and prediction horizon grow. We present a GPU-accelerated direct linear solver for branch MPC formulations in which all trajectories share a single root decision node and evolve independently thereafter. By operating at the linear-algebra level, the solver provides a reusable backend for multiple optimization algorithms whose reduced systems have the required symmetric positive-definite structure. The solver exploits two levels of parallelism: across scenarios and along each prediction horizon. A tailored variable ordering enables horizon-parallel Cholesky factorization while preserving a single root-tail coupling block per scenario in the factor. Numerical experiments demonstrate substantial speedups over state-of-the-art sparse direct solvers, achieving factorization speedups of up to 6.0$\times$ over cuDSS and 27.6$\times$ over eight-thread PARDISO, with triangular solve speedups of up to 3.5$\times$ and 15.8$\times$, respectively.

Under Gaussian sampling, removing location and scale turns the sample into a direction that is uniformly distributed on a sphere of dimension $n-2$, and sample skewness and kurtosis become a cubic-quartic polynomial image of spherical measure. We use this geometry to obtain exact finite-sample results for the Jarque-Bera statistic $\mathrm{JB}_n$. First, for every $n\ge 3$ the largest attainable value of $\mathrm{JB}_n$ is $n\{(n-2)^4+4(n-1)^2\}/\{24(n-1)^2\}$, attained by a single-outlier sample, so the asymptotic $10\%$ and $5\%$ tests have size zero for $n\le 6$ and the $1\%$ test for $n\le 7$; near this maximum the density of $\mathrm{JB}_n$ behaves like an explicit multiple of $(J_n^+-x)^{(n-4)/2}$. Second, passing from residual coordinates to power sums writes the joint density of skewness and kurtosis as an integral of the reciprocal square root of a polynomial discriminant; for every $n\ge 5$ the innermost integral runs over a single interval and is a Lauricella $F_D$ period. Third, this yields explicit laws: an arcsine law for $n=3$, an algebraic joint density for $n=4$, and a single complete elliptic integral, equivalently ${}_2F_1(\tfrac12,\tfrac12;1;\cdot)$, for $n=5$. Residual-coordinate collisions generate the discriminant singularities of the joint law, while stationary points of $\mathrm{JB}_n$ on the residual sphere govern the singularities of its one-dimensional density.

In this paper, we prove the equivalence between coerciveness and the existence of a solution to the generalised Monge-Ampère equations for right-Noetherian polynomials satisfying several assumptions. This includes the case of strictly interlacing polynomials. In particular, we have generalised Chu-Lee's result to supercritical but not hypercritical LYZ equations.

We show that for a 1-category C, the ({\omega}, k)-polygraphic homology of C for any k {\geq} 1, that is taken with cofibrant resolutions in strict ({\omega}, k)- categories, does not depend on k and is canonically isomorphic to the homology of the classifying space of C. When C is a groupoid, we also show this for k = 0. In particular, this means that the classical homology of groups can be obtained by taking cofibrant resolutions in strict {\omega}-groupoids. In order to show these results, we develop the theory of discrete Conduché fibrations in the category of strict ({\omega}, k)-categories, building on previous work by the first-named author.

Many iterative algorithms rely on bootstrapping. A variable is updated using a second, frozen copy as a target, which is periodically replaced with the updated variable. Majorize-minimize and inexact proximal-point methods share this structure, as does temporal-difference (TD) learning. However, existing convergence guarantees for scenarios that combine sampled updates with targets refreshed only every $K$ steps rely on the specific structure of the update, such as linear approximation or gradient-based inner steps, and on uniformly bounded sampling error. We instead model the sampled update as a stochastic operator on the parameter space, which reduces the analysis to a contraction argument that needs no gradient structure and allows the sampling error to grow with the iterates. Within this framework, we derive a finite-time bound for i.i.d. samples and any target-update period $K$. We show that the iterates converge geometrically in root mean square to a ball around the fixed point, provided the sensitivity to the frozen target is smaller than the contraction slack of the inner map. Existing deterministic frozen-target contraction and stochastic-gradient-type bounds follow as special cases of our framework, and simulations of TD learning reproduce the predicted contraction rate and scaling of the error floor with the step size.

We study the Theta operators of D'Adderio-Iraci-Vanden Wyngaerd at $t=1$ and show that the power-sum-indexed operators $\Theta_{\mathsf p_k}|_{t=1}$ agree with a commuting family of derivations when restricted to symmetric functions of positive degree. Writing $\widetilde{h}_a$ for the modified Macdonald function indexed by the single row $(a)$ we establish that $\Theta_{\mathsf p_\mu}\widetilde{h}_a|_{t=1}$ is, up to a normalization, the single-row Macdonald cumulant of Dolęga. We use these results to show that $\Theta_{\mathsf p_\mu}\widetilde{h}_a|_{t=1}$ and $\Theta_{\mathsf e_\lambda}\widetilde{h}_a|_{t=1}$ are both sums of vertical-strip LLT polynomials indexed by certain plane trees. The former further shows that single-row Macdonald cumulants are LLT-positive, thereby yielding a stronger form of the higher-order Macdonald positivity conjecture of Dolęga when all shapes are single rows.

Let $\mathcal G=H\rtimes\Gamma$ be a semidirect product of a finite group $H$ with $\Gamma=\mathbb Z_p^d$, the direct product of finitely many copies of the additive group of the $p$-adic integers. In the case $d=1$, Iwasawa algebras of such profinite groups were studied by Ritter–Weiss, Lau, Nickel, and the author. We generalise these results to all $d\ge1$, describing the Wedderburn decomposition of the total ring of quotients of the Iwasawa algebra of $\mathcal G$.

We study nonlinear logarithmic Hausdorff-Young inequalities with constant $1$ for $\mathrm{SU}(1,1)$-valued Fourier products indexed by binary cubes. The inequalities are proved for the optimal range of exponents, extending into the region $1/p + 1/q < 1$. The proof combines majorization, a known sharp two-point inequality, and tree iteration. We also obtain the corresponding chain inequality and nonlinear bounds for generating functions of alternating chains, recovering the known sharp bounds for additive energies on the binary cube as the amplitudes vanish.

We consider a class of SPDEs driven by a standard real-valued Brownian motion, with drift and diffusion coefficients such that there exists a unique mild solution taking values in the interval $[-1,1]$ almost surely. To preserve this qualitative property of the exact solution, we propose a domain preserving Lie–Trotter splitting scheme: for any choice of the time-step size, the numerical solution takes values in the interval $[-1,1]$ almost surely. Furthermore, we prove mean-square convergence with rate $1/2-$ for the domain preserving Lie–Trotter scheme. These theoretical results are illustrated with numerical experiments.

We present an analytic algorithm to compute isolated values of the divisor summatory function in $N^{9/28+o(1)}$ expected time, improving upon existing methods that require $O(N^{1/3})$ time. Our method significantly departs from elementary approaches to analyze the geometry of the Dirichlet hyperbola, relying instead on Voronoï summation and accelerating the evaluation of the resulting exponential sum.

We show that the evaluations of combinatorial 1-cocycles defined by Gauss diagrams with a triangle on the canonical loops in the space of long knots are finite type invariants. Using Gauss diagrams, we construct two $\mathbb{Z}$-valued combinatorial 1-cocycles $\beta_1, \beta_2$ and a $\mathbb{Z}/2\mathbb{Z}$-valued 1-cocycle $\beta_3$ on the space of long knots and prove their cocyclicity by verifying their invariance under higher Reidemeister moves coming from the codimension-two singularities of plane curves. We show that they represent genuinely new 1-cohomology classes and compute their pairings with the rotation, rolling, half rolling, bracket and half bracket loops. A key new feature is that $\beta_1$ and $\beta_2$ can pair nontrivially with bracket and half-bracket loops. We conjecture that $(\alpha_3^1,\beta_1,\beta_2)$ over $\mathbb{Q}$, and their mod 2 reductions together with $\beta_3$ over $\mathbb{Z}/2\mathbb{Z}$, form bases of degree-one cohomology up to order 4 in the sense of Vassiliev. In addition, we show that the reparametrization loop is homotopic to the rolling loop concatenated with the rotation loop in $\operatorname{Emb}(S^1, S^3)$. Finally, we give the criteria for a 1-cohomology class in the long knot space to descend to 1-cohomology classes in $\operatorname{Emb}(S^1, S^3)$ and $\operatorname{Emb}(S^1, S^3)/\operatorname{Diff}^{+}(S^1)$. Using these criteria, we show that $\beta_1$ descends to a nontrivial 1-cohomology class in $\operatorname{Emb}(S^1,S^3)$, while $\beta_1 \bmod 2$ and $\beta_3$ descend to linearly independent nontrivial 1-cohomology classes in $\operatorname{Emb}(S^1, S^3)/\operatorname{Diff}^{+}(S^1)$.

Carathéodory's construction, also known as Method I, turns any weight on a family of sets into an outer measure. The difficulty is the extension problem: showing that the outer measure retains the prescribed weights. We record elementary properties of Method I, beginning with the fact that countable subadditivity on the covering family is exactly the criterion for faithful extension, and use them to prove standard results with only outer measures, $\sigma$-algebras, and measures, without premeasures on algebras. In topological spaces, a continuity principle reduces this criterion to finite subadditivity and finite approximation by open and compact sets. Applications include the Lebesgue integral as a measure, Tonelli's theorem, multidimensional Lebesgue–Stieltjes measures, Riesz representation for vector-valued functionals, Kolmogorov extension, mass distributions on nested partitions, and Frostman's lemma.

We prove that, whenever $ p \ge n^{-1/2 + o(1)} $, with high probability $ G(n, p) $ admits a fractional triangle decomposition, that is, a non-negative weight function on its triangles for which the total weight of all triangles containing each edge is equal to 1. This bound on $ p $ is optimal up to the asymptotic error term, improving upon the recent state of the art, due to Mahabaduge and Simkin, that $ p \ge n^{-4/11 + o(1)} $ suffices. Our main tool is a deterministic theorem guaranteeing the existence of fractional clique decompositions in all hypergraphs satisfying suitable `clique-regularity' properties. We prove this by analysing an extension (and generalisation to hypergraphs) of an algorithm proposed by Mahabaduge and Simkin, in which, at each time step, the discrepancy at each edge is spread among its containing triangles. By showing the concentration of the relevant quantities in random $ k $-uniform hypergraphs, we obtain for all $ k \ge 2 $ and $ r \ge k + 1 $ that w.h.p. $ G^{(k)}(n, p) $ admits a fractional $ K^{(k)}_r $-decomposition whenever $ p \ge n^{-\frac{r - k}{\binom{r}{k} - 1} + o(1)} $, which improves upon results of Delcourt, Kelly, and Postle, and is best possible up to subpolynomial factors.

Highest weight categories with finitely many simples correspond to quasi-hereditary algebras. Ringel duality shows that, up to Morita equivalence, quasi-hereditary algebras come in pairs, while Ringel self-duality is the phenomenon in which a quasi-hereditary algebra is paired with itself. The main purpose of this paper is to show that there are connected non-semisimple hereditary algebras that are Ringel self-dual and to give a combinatorial interpretation of Ringel self-duality in terms of Auslander-Reiten theory. Let $A$ be a basic connected hereditary algebra over an algebraically closed field $k$ equipped with a quasi-hereditary structure. We show that, if $A$ is of finite representation type and Ringel self-dual, then the indecomposable summands of the characteristic tilting module $T$ are obtained from the indecomposable projectives, and from indecomposable injectives by applying powers of the Auslander-Reiten translation, in a way governed by an automorphism of the underlying Dynkin diagram. In particular, the number of indecomposable $A$-modules has the same parity as the number of simple $A$-modules. Conversely, we show that if $T\cong \tau^{-n}(A)\cong \tau^n\Hom_k(A, k)$ for some natural number $n$, then $A$ is Ringel self-dual and of finite representation type. Using these results, we determine the simply laced Dynkin diagrams admitting an orientation and a quasi-hereditary structure for which the corresponding path algebra is Ringel self-dual: they are exactly those of type $D_{2m}$ with $m\geq 2$ and $A_{4t+1}$ with $t\geq 0$. In particular, no exceptional type occurs, and no connected non-semisimple hereditary Nakayama algebra is Ringel self-dual.

Many real-world optimization problems rely on expensive simulations or experiments, making the efficient use of available data essential. Multi-fidelity optimization of high-dimensional black-box functions subject to black-box constraints is increasingly relevant as the cost of objective evaluations continues to rise in applications such as machine learning, engineering, and control. To our knowledge, no existing method simultaneously addresses high-dimensionality, black-box constraints, an arbitrary number of fidelity levels, and non-nested sampling. In this work, we extend the Scalable Constrained Bayesian Optimization method to the multi-fidelity setting, resulting in the MF-SCBO method. The proposed approach is evaluated on standard benchmark functions as well as challenging problems. The experimental results demonstrate that MF-SCBO generally achieves better convergence than both the single-fidelity SCBO and the other multi-fidelity method considered in this high-dimensional and constrained settings.

In this paper, we resolve the problem of finding the required level of truncation needed on the isomorphism class of a truncated valuation ring of a complete discretely valued field of mixed characteristic to determine the isomorphism class of the field itself.

Billera, Jia, and Reiner (2009) introduced the quasisymmetric functions of matroids and showed that this defines a Hopf algebra homomorphism which is a valuative invariant, i.e., isomorphic matroids have the same quasisymmetric function and polytopal subdivisions of matroid base polytopes define relations among the corresponding quasisymmetric functions. In this project we study an analogue in non-commuting variables, the word-quasisymmetric functions. To every matroid $M$ we associate a word-quasisymmetric function $\psi(M)$ and call this the chromatic word-quasisymmetric functions of a matroid. Matroids and word-quasisymmetric functions form Hopf algebras, and our map $\psi$ between them is a homomorphism. We want to study the kernel, equivalently the image, of the map $\psi$ from matroids to word-quasisymmetric functions, that is, we would like to understand which matroids are indistinguishable by the chromatic word-quasisymmetric functions. The map $\psi$ is not an invariant, but we can show that it is valuative. Using Schubert matroids and nested matroids, special classes of matroids, we prove a lower bound of $2^d-d$ for the rank of the map $\psi$ from matroids to the chromatic word-quasisymmetric functions in degree $d$ and conjecture the upper bound of $d!$ is tight.

Given two (closed) subgroups H, K of a profinite group G, we write Pr(H,K) for the probability that a random pair from HxK commutes. Here we are concerned with profinite groups containing Sylow subgroups P, Q such that Pr(P,Q) > 0. First, we show that if G is a profinite group containing a Sylow 2-subgroup L and a Sylow p-subgroup P , where p is odd, such that Pr(L,P) is positive, then G is virtually pro-p-soluble (Theorem 1.1). Then we handle similar issues for profinite groups admitting coprime automorphisms. In particular, we prove that if G is a profinite group admitting a group of coprime automorphisms A such that there is a Sylow 2-subgroup L of C_G(A) and an A-invariant Sylow p-subgroup P of G, where p is odd, for which Pr(L,P) > 0, then G is virtually pro-p-soluble (Theorem 1.3). On the other hand, if we only have Pr([L,A],[P,A]) > 0, then [G,A] need not be virtually pro-p-soluble. We show that in this case [G,A] has an open normal subgroup of non-p-soluble length at most 1 (Theorem 1.4). Furthermore, if P is an A-invariant Sylow p-subgroup of G such that Pr([P, A],[P, A]^x) > 0 for every x in G, then [G,A] has an open normal subgroup of non-p-soluble length at most 1 (Theorem 1.5).

We construct an explicit smooth projective Fano fourfold $X$ over an algebraically closed field $k$ of characteristic two with $H^1(X,\mathcal{O}_X)\simeq k$. Moreover, $X$ does not admit any free rational curve, and so it is not separably uniruled. The example has been constructed using ChatGPT.

We prove that the optimal Lojasiewicz–Simon exponent of a real-analytic functional on a Banach space is an intrinsic invariant of its finite-dimensional Lyapunov–Schmidt singularity under natural Fredholm and pairing-compatible hypotheses. More precisely, Lyapunov–Schmidt reduction preserves the entire set of admissible exponents, while the reduced germ on the kernel of the Hessian is intrinsic up to analytic right-equivalence tangent to the identity. We then identify this invariant with a real relative Lojasiewicz exponent of the pair consisting of the Jacobian ideal of the reduced energy and its principal energy ideal. Consequently, the optimal exponent admits equivalent descriptions by analytic arcs and by a finite maximum of divisorial ratios; in particular, it is rational. In Newton-nondegenerate cases, these formulas yield explicit monomial expressions. We apply the theory to several classes of singular energies. For the sum $Q_k$ of the squares of all $k\times k$ minors of a real matrix, we determine the associated ideal pair up to real integral closure and obtain the exact local exponent $$1-\frac1{2(k-s)}$$ at matrices of rank $s<k$. For Yang–Mills energy, we prove that the product flat connection on $\mathbb T^2$ has optimal exponent $3/4$ for every nonabelian compact structure group. For $\mathrm{SU}(2)$, the determinantal reduction extends this value to product flat connections on every $\mathbb T^d$, while sufficiently small nonzero constant flat connections have exponent $1/2$. Finally, for resonant semilinear Dirichlet energies, we identify higher-order resonant obstructions, establish an all-order obstruction ladder for simple resonance, and obtain a complete parity-dependent classification on $(0,\pi)$.

We establish a stability threshold for the 3D Boussinesq equations around Couette flow $(y,0,0)$ with linearly stratified temperature profile $1+y$ on $\mathbb{T}\times\mathbb{R}^2$, with distinct viscosity $\kappa$ and thermal diffusivity $\mu$ and arbitrary Brunt-Väisälä frequency $\beta>0$. Assume that $\mu$ and $\kappa$ are comparable, and set $\nu=\min\{\kappa,\mu\}$. For sufficiently small \(a_0>0\) and \(s\geq4\), we prove that if the initial perturbation satisfies $\|(u_{in},\theta_{in})\|_{H^s}+\|(u_{0,in},\theta_{0,in})\|_{W^{s,1}}\lesssim\nu^{\frac{2}{3}} |\ln{\nu}|^{-2-2a_0}$, then the corresponding solution exists globally in time. We also establish Strichartz-type estimates for zero mode. Moreover, our estimates further allow for a nonlinear amplification of order $\nu^{-1/6}$ in the high-regularity norm of the zero mode of the first component $u^1_0$.

This letter investigates a non-uniform activation aided distributed generalized spatial modulation (DGSM) scheme in a downlink distributed multiple-input multiple-output (DMIMO) system formed by geographically separated transmission reception points (TRPs). Unlike conventional uniform-activation DGSM, the proposed scheme assigns adaptive activation probabilities to different TRP subsets, thereby jointly exploiting the data-domain transmission capability and the index-domain information carried by cooperative subset selection. We derive a mutual-information upper bound under non-uniform activation and formulate a joint activation-probability and power allocation problem subject to a long-term average power constraint. To solve this problem, we develop a semi-closed-form joint probability and power optimization (SC-JPPO) algorithm, where the optimal activation probabilities are shown to follow a Softmax function of the effective utilities. Simulation results demonstrate that the proposed scheme improves spectral efficiency under the same activation scale and power budget compared with existing benchmarks.

In this paper, we establish a general identity for three-term recurrence sequences and then give applications to orthogonal polynomials and Apéry-like numbers.

For a $3$-graph $F$, the Turán number of $F$, denoted by $\ex(n,F)$, is the maximum number of edges in a $3$-graph on $n$ vertices containing no subgraph isomorphic to $F$. Let $F^-_{4,3}$ be the $3$-graph formed by a complete four-vertex core and three outer vertices, with all but one of the twelve triples containing one core vertex and two outer vertices. We prove that, for every $n\ge8$, \[ \ex(n,F^-_{4,3})=\binom n3-\binom{\lfloor n/2\rfloor}{3}-\binom{\lceil n/2\rceil}{3}, \] and the balanced complete bipartite $3$-graph is the unique extremal configuration. This determines the exact value and all equality cases in the asymptotic theorem of Mubayi and Rödl. It also extends the exact Turán Number of $F_{3,3}$ and resolves a conjecture of Frankl, Huang and Rödl.

Yang introduced a tempered xi function $\widehat\xi(s)$ by replacing the hyperbolic cosine in a classical integral representation of the Riemann xi function by a hyperbolic sine, and conjectured that every zero of $\widehat\xi$ lies on the critical line $\Re s=1/2$. We disprove this conjecture. After the change of variables $x=e^{2t}$, one has \[ \widehat\xi\!\left(\tfrac12+iz\right)=iS(z), \qquad S(z)=\int_0^\infty K(t)\sin(zt)\,dt, \] where $K$ is positive, smooth, and doubly exponentially decaying. Two integrations by parts give $S(y)=K(0)/y+O(y^{-2})$ for real $|y|\to\infty$, with $K(0)>0$; hence $S$ has only finitely many real zeros. On the other hand $S$ is an entire function of order at most one. If it had only finitely many zeros in the complex plane, Hadamard factorization would force $S(z)=P(z)e^{az+b}$, which is incompatible with $S(y)\to0$ as $y\to\pm\infty$. Thus $S$ has infinitely many nonreal zeros, and consequently $\widehat\xi$ has infinitely many zeros off the critical line. The proof is unconditional and does not use the Riemann Hypothesis or numerical zero finding. A short numerical illustration is included.

We study the Schatten class membership and essential norms of the operators $C_\varphi C_\psi^*$ and $C_\psi^*C_\varphi$, where $C_\varphi$ and $C_\psi$ are composition operators with nonconstant linear fractional symbols, acting on weighted Dirichlet spaces $\mathcal{D}_\alpha$ with $\alpha>-1$. We obtain complete characterizations of their Schatten class membership in terms of the boundary behavior of $\varphi$ and $\psi$. In particular, the corresponding geometric conditions are independent of the Schatten exponent, and each product belongs to every Schatten class whenever its boundary condition is satisfied. We also investigate the essential norms of these products. For $\alpha=0$ and $\alpha>0$, we obtain exact formulas in terms of boundary contact and the derivatives of the symbols, while for $-1<\alpha<0$ we obtain two-sided estimates by reducing the problem to positive Toeplitz operators and local averages of generalized Nevanlinna counting functions.

The speed v(lambda) of the lambda-biased random walk on a supercritical Galton-Watson tree without leaves is conjectured to be nonincreasing on [0,m), where m is the mean offspring. For non-constant offspring, monotonicity is known only for small bias: lambda <= 1/1160, lambda <= 1/2, and, when every vertex has at least m_1 >= 2 children, lambda <= m_1/(1+sqrt(1-1/m_1)). For offspring uniform on {2,3} (m=2.5) the last bound is 2/(1+sqrt(1/2)) = 1.17157... We prove, with computer assistance, that v is strictly decreasing on [0,1.755] for this law. The proof has three parts. Aidekon's speed formula gives v=(R-lambda)/(R+lambda) for an explicit functional R, so v decreases exactly when R/lambda does; we compare R/lambda at two biases directly, which avoids differentiating the conductance. A pathwise Lipschitz bound on the conductance in lambda turns that comparison into an inequality between expectations of explicit functions. A monotone sandwich of discretised laws gives two-sided bounds on the conductance law, and each lambda-cell is verified with exact rational arithmetic on top of bounded floating-point error; an independent interval-arithmetic implementation re-certifies five cells, including the one that sets the endpoint. The method stops where the crude Lipschitz bound becomes too weak; sharper control of the derivative of the conductance is what the full range needs. Code and certificates are public.

The classical Becker-Döring equations describe the formation of clusters by aggregation and fragmentation of monomers. If the total amount of mass is supercritical, larger and larger clusters are formed, leading to an asymptotic loss of mass in the long time limit. The two-component Becker-Döring system arises when clusters are built from two types of monomers. Here, we study the natural extension of the one-component system, where no energy or entropy is leaving or entering the system - the so called detailed balance assumption. We rigorously prove that all initial conditions admit a solution minimising the relative entropy as time approaches infinity. This shows, that the long time limit in the weak* topology is selected through the initial Type I and Type II masses. Furthermore, the proportion of lost Type I mass is determined by the limit point via the mixing ratio of Type I and Type II monomers that maximises the binding energy. The main difficulty of the two-component system is that no relative entropy is weak* continuous, which is crucial for the classical argument [2]. Instead, our proof is based on a discrete two-dimensional logarithmic Sobolev inequality, that bounds the entropy dissipation on the correct timescale. Our approach also improves current assumptions to determine the long time behaviour for the one-component system.

We develop randomized branch methods for finding Bouligand-stationary (B-stationary) points of linear and quadratic programs with complementarity constraints (LPCCs and QPCCs) using only linear programming subproblems. The methods exploit the finite-union geometry of the feasible set and search for first-order descent on randomly selected compatible branches. For LPCCs, an exact method optimizes over sampled branches, while an inexact simplex method can accept an improving branch-feasible vertex before solving the sampled penalty problem to optimality. For QPCCs, including problems with indefinite quadratic objectives, branch quadratic programs are replaced by linearized trust-region subproblems; a ratio test ensures actual decrease, and thresholded sampling detects branches that emerge only at accumulation points. Under the stated assumptions, the LPCC methods stabilize after finitely many changes at B-stationary points almost surely. For the QPCC method, finite termination yields a B-stationary point, and every accumulation point of an infinite run is B-stationary almost surely. Experiments on bilevel-induced instances, instances arising from inverse quadratic programming, and sparse affine generalized Nash equilibrium instances, together with 129 MacMPEC embedding tests, show that the methods return points with competitive objective quality and runtimes on large-scale complementarity systems.