Papers

We study a class of congestion games with a continuum of heterogeneous players and a finite set of resources, without assuming the existence of a potential function. Players differ in their individual characteristics and may contribute differently to congestion, which is represented through general aggregate features of the population choosing each resource. We characterize the conditions under which such a game admits a potential and show that this structure generally fails when players have heterogeneous impacts on congestion. To analyze the non-potential case, we introduce the concept of Stochastic Variational Nash Equilibrium, in which players observe idiosyncratic cost shocks before selecting a resource. We show that equilibrium choices are pure conditional on players' characteristics and shock realizations, whereas the induced aggregate allocation is probabilistic. Under Gumbel-distributed shocks, the equilibrium is equivalent to a continuum-player logit Quantal Response Equilibrium. Using a fixed-point formulation and the Schauder–Tychonoff theorem, we establish existence for every positive noise level and recover a pure Nash equilibrium in the zero-noise limit. Under a coordinate-wise monotonicity condition on aggregate interaction costs, we further prove uniqueness and convergence of the regularized allocations to the pure equilibrium. An application to electric vehicle charging illustrates how heterogeneous energy demands jointly affect station choice, congestion, and market shares.

We give an estimate for the distribution tail of the passage time between $0$ and $\infty$ in first-passage percolation on $\mathbb Z^d$, when the weight distribution is bounded and has a supercritical atom at $0$. A special case involves the number of disjoint cutsets between $0$ and $\infty$ in supercritical bond percolation. Besides, we show that the rate function for the upper-tail deviations of the point-to-point passage time is asymptotically polynomial near $0$.

This paper studies the spectral compressed sensing problem, which aims to recover an $r$-spectrally-sparse signal of length $N$ from linear compressive measurements. We reformulate the problem as the recovery of a low-rank Hankel matrix. Motivated by this formulation, we propose a Lifted Restricted Isometry Property (LRIP) for the measurement matrix and show that Gaussian measurement matrices satisfy LRIP under certain conditions. Unlike classical RIP, LRIP imposes a rank constraint in the lifted matrix domain while taking measurements in the signal domain through the adjoint Hankel operator. To solve the resulting nonconvex problem efficiently, we introduce a Fast Iterative Hard Thresholding (FIHT) algorithm. Based on LRIP, we show that FIHT converges linearly to the true signal with high probability using $\mathcal{O}(r \ln r \ln^2 N)$ scaled Gaussian measurements, which is near-optimal up to log factors and improves the state-of-the-art order of $r^2$ for nonconvex Hankel methods. Extensive numerical results demonstrate the superiority of FIHT as compared to state-of-the-art convex methods in terms of phase transitions, computational time, and robustness to noise.

We study the minimum-norm Gauss–Newton method for phase retrieval from random coded diffraction patterns with additive intensity noise. For bounded symmetric masks satisfying the standard complex moment conditions, we establish convergence guarantees with quadratic error recursions in both the resampled and fixed-data settings. In the resampled setting, $O(\log^2 n)$ masks per independent block suffice to obtain a spectral initializer within a constant-relative-radius neighborhood and to establish a local quadratic error recursion with an additive noise term. Consequently, in the noiseless case, the method achieves relative accuracy $\epsilon$ using $O(\log\log(1/\epsilon)\log^2 n)$ masks. In the fixed-data setting, $O(\log^6(en))$ masks suffice for spectral initialization within a neighborhood whose relative radius is proportional to $\log^{-5/2}(nL)$. Within this neighborhood, the method satisfies a linear contraction bound and eventually a quadratic error recursion. In particular, in the noiseless case, the method initially converges linearly and enters a quadratic convergence phase after $O(\log n)$ iterations, without acquiring additional masks. Numerical experiments on synthetic signals and a natural image demonstrate the method's efficiency relative to state-of-the-art algorithms.

We study two inclusion-wise extremal variants of the 0-1 Knapsack Problem (KP). Given $n$ items with integer weights and nonnegative costs or profits, the Minimum-Cost Maximal Knapsack Packing Problem seeks a minimum-cost packing for capacity $c$ to which no item can be added, while the Maximum-Profit Minimal Knapsack Cover Problem seeks a maximum-profit cover for threshold $d$ from which no item can be removed. Both problems are weakly $\mathrm{NP}$-hard and pseudo-polynomially solvable, as is the classical KP, but their approximability is fundamentally different: unless $\mathrm{P}=\mathrm{NP}$, with positive values neither problem admits a polynomial-time $2^{p(n)}$-approximation for any polynomial $p$; allowing zero values rules out every multiplicative factor. Even with unit costs for packing and unit profits for covering, neither admits an $n^{1-\varepsilon}$-approximation for any fixed $\varepsilon>0$, whereas a trivial $n$-approximation exists. We identify restrictions that restore approximability. If every item weighs at least a fixed fraction of the capacity, respectively of the threshold, both problems can be solved exactly in polynomial time. For the cover problem, if each weight $w_i$ satisfies $w_i-1\ge\delta(d-1)$, we give an approximation scheme for arbitrary profits whose running time is polynomial in $n$, $1/\varepsilon$, and $1/\delta$. If values equal weights, a fully polynomial-time approximation scheme (FPTAS) was already known for the packing problem; we give an FPTAS for the cover problem, improving on the previously known factor $2$. If value-to-weight ratios differ by at most a factor $\kappa$, we obtain a $(1+\varepsilon)\kappa$-approximation for both problems.

In this paper, we investigate the family of $H^{*}$-open sets from the perspective of generalized topology and examine the structural and separation properties arising from it. For a topological space $(\mathbb{X},\tau)$, the family of all $H^{*}$-open subsets of $\mathbb{X}$ is considered as a natural candidate for an $H^{*}$-generated generalized topology. We establish conditions under which this family forms a strong generalized topology and identify additional assumptions that allow the resulting structure to be treated as an ordinary topology. This distinction provides a systematic framework for studying $H^{*}$-open sets without imposing finite-intersection properties in advance. Within this framework, we investigate the corresponding $H^{*}$-interior and $H^{*}$-closure operators and analyze separation properties determined by the $H^{*}$-open structure. When the $H^{*}$-generated family forms a topology, the resulting associated space is used to relate generalized $H^{*}$-separation properties to their classical topological counterparts. We further examine compactness-type properties, subspace behaviour, and preservation under suitable mappings. Examples are included to illustrate how the structure generated by $H^{*}$-open sets may exhibit separation behaviour different from that of the original topology. The results provide a unified setting for connecting $H^{*}$-open-set theory with generalized topological structures and offer a basis for further investigation of separation and covering properties in this framework.

In this article, we study optimal control problems driven by mean-field forward-backward stochastic differential equations (FBSDEs). The aim is two-fold: (i) to provide the unique optimal control; (ii) to establish classical solutions of the associated generalized Hamilton-Jacobi-Bellman (HJB) equations on the space of probability measures. For (i), we study the coupled system arising from the maximum principle, which involves both the state FBSDEs and the adjoint FBSDEs. This system is non-standard and more complicated than that in standard mean-field type control (MFTC) problems. By extending the method of continuation in the coefficients to our framework, we establish the well-posedness result, and subsequently obtain the differentiability of the value functional with respect to the distribution argument — thereby extending existing results for MFTC problems. For (ii), since the flow property does not generally hold for control problems of FBSDEs, standard dynamic-programming arguments cannot be applied directly. We therefore develop probabilistic approaches to the classical solvability of generalized HJB equations under three distinct sets of conditions. The auxiliary processes arising in these arguments differ significantly from those in standard MFTC problems. We establish new connections between the value function and the adjoint processes, derive equations for the relevant decoupling field, and obtain first- and second-order generalized HJB equations, including a coupled system on the space of probability measures. The standard HJB equations for MFTC problems are recovered as special cases. Our approach yields classical solutions without requiring non-degeneracy of the diffusion.

We study the eigenvalues of the negative Dirichlet Laplacian $-\Delta$ in domain $D_{\epsilon,N}$ obtained by removing $N$ Euclidean balls of radius $\epsilon$ from a bounded open set $D\subseteq\mathbb R^d$ ($d\ge2$). It is known that, for the balls centered at points drawn independently from a distribution with probability density $\rho$ and for $N\to\infty$ and $\epsilon\downarrow0$ so that the aggregate Newtonian capacity of the removed balls tends to some $\kappa>0$, the eigenvalues of $-\Delta$ in $D_{\epsilon,N}$ homogenize to those of the Schrödinger operator $-\Delta+\kappa\rho$ in $D$. Here we prove that, for the eigenvalues that are simple in the limit, a multivariate Central Limit Theorem holds in all $d\ge2$ under centering by expected values. Unlike earlier work in $d=3$ by Figari, Orlandi and Teta and, independently, Ozawa from 1980s that was based on resolvent expansions, we rely on martingale representations and one-ball perturbation asymptotic controlled by methods of harmonic analysis.

For every von Neumann algebra \(\M\), we prove that each bounded sequence in the Haagerup space \(L_1(\M)=\M_*\) admits either a norm-convergent subsequence or a \(2\)-co-lacunary subsequence. This answers an open problem of Parcet and Randrianantoanina \cite[Remark~3.11]{ParcetRandrianantoanina2006} and removes both hyperfiniteness and semifiniteness.

We study minimizers of the optimal Sobolev quotients for the critical embeddings $D^{1,p}(\mathbb R^N;\mathbb R^m)\hookrightarrow L^{p^*}(\mathbb R^N;\mathbb R^m)$ and $W^{1,p}(\Omega;\mathbb R^m)\hookrightarrow L^{p^*}(\Omega;\mathbb R^m)$. In the whole space, we classify all minimizers as $\boldsymbol u=\boldsymbol c\,U_{\delta,x_0}$, where $\boldsymbol c\neq\boldsymbol0$ and $U_{\delta,x_0}$ is an Aubin–Talenti bubble. On bounded domains and for $\lambda>0$, we equip $W^{1,p}(\Omega;\mathbb R^m)$ with $ \|\boldsymbol u\|_\lambda^p :=\int_\Omega \bigl(|D\boldsymbol u|^p+\lambda|\boldsymbol u|^p\bigr) $. Using scalar attainment results, we establish existence and synchronization of vectorial minimizers. At resonance, we consider the pure Neumann problem $ -\boldsymbol\Delta_p\boldsymbol u =|\boldsymbol u|^{p^*-2}\boldsymbol u\ \text{in }\Omega. $ For the associated quotient, we prove existence and global $C^1(\overline\Omega)$ regularity of minimizers. Finally, we prove synchronization of radial solutions and deduce symmetry breaking on balls.

We study the two-dimensional non-rotating primitive equations with anisotropic viscosity and no thermal diffusion. For doubly periodic perturbations of a hydrostatic equilibrium with an affine temperature profile, we establish local Hadamard well-posedness in Sobolev spaces under strict stratification. The initial horizontal velocity and temperature have sufficient regularity and vanishing vertical means, but need not be small. The estimates close at the same Sobolev order for these variables despite the horizontal derivative loss in the diagnostic vertical velocity. Stratification-weighted energy estimates and a horizontal compensation argument avoid an additional temperature derivative while retaining the full vertical dependence and the vertically homogeneous velocity component determined by periodic compatibility. A one-sided comparison places higher-order norms of the more regular solution only in terms involving lower-order differences. This structure allows the Bona–Smith method to yield continuous dependence in the original Sobolev topology. We obtain a unique maximal strong solution in the strictly stratified class, together with continuous dependence of the diagnostic vertical velocity and pressure in their natural Sobolev spaces and lower semicontinuity of the maximal lifespan.

Classical Turán theory relates the asymptotic extremal problem for an ordinary forbidden graph to its chromatic number. This principle fails when a forbidden copy must respect prescribed vertex colors, as occurs naturally in multipartite extremal problems and in embeddings with prescribed clusters. We study properly vertex-colored graphs whose hosts have a prescribed density between every pair of color classes, with containment interpreted through color-preserving homomorphisms. Our main theorem gives a finite directed reduction: for every fixed $m$ and $\varepsilon>0$, every sufficiently many-colored $d$-dense host admits a colored homomorphism from an $m$-layer half-blowup $D[m]$ of a finite digraph $D$ with density at least $d-\varepsilon$, where $D$ may be chosen with $\lceil\varepsilon^{-1}\rceil$ vertices. Consequently, the colored Turán density of every non-empty finite forbidden family equals the homomorphism Turán density of its associated acyclic orientations. We then determine the complete finite value set: these densities are exactly the Lagrangians of finite loopless digraphs, and hence are rational. The main technical input is a low-resolution sampling method that preserves all pairwise bipartite degree profiles on a large subfamily of color classes while keeping the sample size independent of the number of retained classes.

We use Loewner-Kufarev evolution to decompose the Brownian loop soup into a Poisson point process on Brownian bubbles which we call the Brownian bubble tea driven by $\rho$, where $\rho$ is the driving measure for the Loewner chain. The only assumption on $\rho$ is that the evolving boundaries are locally connected. This generalizes a similar disintegration result of Lawler and Werner to essentially optimal regularity, and includes foliations with finite Loewner-Kufarev energy as well as radial and whole-plane $\mathrm{SLE}_\kappa$ for all $\kappa \geqslant 0$. As applications, we give Brownian loop interpretations of the conformal distortion formula for Loewner-Kufarev energy and the relationships between the Loewner energy, Loewner-Kufarev energy, and the Schwarzian action.

We give two counterexamples to Stapledon's Effectiveness Conjecture. Each counterexample is a $6$-dimensional lattice polytope with an order-two symmetry whose equivariant $H^*$-series $\varphi[t]$ is a polynomial with one coefficient that is not effective. For polytopes with a symmetry of order two, the goal of this paper is to build up a picture of the landscape of results for when the Effectiveness Conjecture holds. Our main result is a criterion, in terms of the faces of $P$, that determines when the Ehrhart ring of $P$ admits an invariant linear system of parameters of the correct representation type. We use these results to reduce the space of possible counterexamples to one of two cases. A randomised computer search produced counterexamples in each case. We also give an example where the equivariant $H^*$-series is effective but admits no invariant linear system of parameters, neither for $P$ nor any invariant subdivision of it, answering a question of Stapledon in the case of groups of order two.

We prove that the rational Chow ring of every connected component of the multi-scale compactifications of the genus three strata $\mathcal{H}(4), \mathcal{H}(2,2)$ and $\mathcal{H}(3,1)$ is generated by boundary-stratum classes. Our proof shows Chow-Künneth generation property of boundary strata of the multi-scale compactification by reducing the boundary analysis to lower-genus generalized strata of meromorphic differentials, including residueless loci, described through plane-quartic and plane-cubic models. As a consequence, the rational cycle-class map is an isomorphism and the compactifications have no odd rational cohomology.

Immersed-boundary methods avoid body-fitted meshes, but the energy or entropy stability of the underlying flow discretisation does not automatically extend to the immersed treatment. This manuscript analyses the volume-penalised compressible Navier–Stokes equations for both energy and entropy stability, on the continuous and semi-discrete levels, and integrates the isolated source exactly in time. The source is formulated as a one-parameter family indexed by $\theta$, whose momentum component relaxes the fluid velocity towards the prescribed solid velocity; $\theta$ sets how much of the resulting kinetic-energy loss leaves the domain rather than being converted to internal energy. The two stability requirements impose opposing bounds on $\theta$, with dissipation of the relative total energy requiring $\theta\geq0$ and stability of the mathematical entropy requiring $\theta\leq1$. Where penalisation acts, both conditions are therefore satisfied if and only if $0\leq\theta\leq1$, and the symmetric value $\theta=1/2$ divides the kinetic-energy loss equally between the domain and the internal energy. We conclude that any entropy-stable split-form discontinuous Galerkin spectral element method remains semi-discretely entropy stable once the source is added at the collocation nodes, for stationary and prescribed moving bodies alike.

The semantic entropy of a language model is the entropy of the meanings of its sampled answers. The meanings are never observed, and the entropy is estimated from the verdicts of a pairwise equivalence judge. Given the meanings of $n$ answers, a known channel declares each pair equivalent independently, with probability $q+\Delta$ within a meaning and $q$ across meanings. At most $K$ meanings occur, and $q$ is bounded away from $0$ and $1$. For a weak judge, with $n\Delta^2\to0$, the global minimax rate is set by distributions near a vertex of the simplex and overstates the difficulty elsewhere. We show that the difficulty is set by how many meanings can be rare. Over distributions with at most $s$ meanings below a fixed probability $\tau<1/(K-s)$, the minimax squared risk is of order $N^{-1/s}$ for $s\ge2$, $N^{-1}\log^2(e+N)$ for $s=1$ and $N^{-1}$ for $s=0$, capped at one, where $N=n^K\Delta^{2K-2}$. The matching lower bounds hold on every fixed polynomial window $n\Delta^2\le n^{-\kappa}$. One estimator attains these rates without knowing $s$ or $\tau$: it averages products of centered verdicts along all spanning trees of every $k$-set of answers, projects the resulting power sums onto the valid moment set and inverts them. The case $s=K-1$ gives the global rate $N^{-1/(K-1)}$ and the consistency threshold: with $\Delta=n^{-\gamma}$ the entropy is uniformly consistently estimable exactly when $\gamma<K/(2K-2)$. On polynomial windows, the same estimator attains the sharp local minimax constant on shrinking neighborhoods of interior points with distinct probabilities. Restricting the average to stars multiplies the constant by $K^{K-3}$.

This paper shows how a Schrodinger bridge (SB) structure emerges from maximum-entropy (MaxEnt) distribution steering of discrete-time nonlinear systems. We also establish that the resulting bridge can be realized by an optimal control policy. We consider general state and input costs, non-Gaussian endpoint distributions, and input constraints. Assuming the existence of a feasible policy with finite objective value, we show that the state-path measure under a MaxEnt optimal control policy is the unique SB. The reference measure of this SB is induced by an optimal policy of the corresponding terminal-unconstrained problem. Moreover, we give an equivalent characterization of the optimal state-path measure using a killed reference and its normalized Feynman–Kac measure, thereby interpreting the state cost as killing. We then characterize the finite-cost feasibility by the existence of an endpoint coupling having finite Kullback–Leibler divergence from the reference endpoint distribution. For compactly supported marginals with bounded target density, we derive reachability-based sufficient conditions guaranteeing this feasibility. We further provide specializations to fully actuated control-affine and reachable linear systems.

We compute Gromov–Witten (GW) and Donaldson-Thomas (DT) invariants (including descendant invariants) for local CY $4$-folds over a smooth quadric threefold $Q$, up to degree $4$. We use the virtual localization formula with respect to a two-dimensional torus action to compute DT invariants. Combining these results with the computation of the J-function of the variety $Q$, we confirm the correspondence between DT and Gopakumar–Vafa invariants suggested by Cao–Maulik–Toda in genus zero. Based on these results, we predict a genus one BPS invariant on $Q$.

In this paper, we first construct infinitesimal equivariant analytic torsion forms for Kähler fibrations. We then compare them with Ma's equivariant analytic torsion forms, modulo $\partial$- and $\overline{\partial}$-exact forms, in terms of equivariant Bott–Chern currents and the $I$-genus. Our comparison formula extends that of Bismut and Goette to families.

We establish a generating function identity for finite and symmetric multiple zeta values using Dougall's bilateral ${}_2H_2$ summation formula. As an application, we prove a restricted sum formula in which the parity of each component of the index is prescribed. We also derive a generating function identity for finite multiple zeta values $\zeta^{}_{\mathcal{A}}(\{1,2\}^n)$ and symmetric multiple zeta values $\zeta^{}_{\mathcal{S}}(\{1,2\}^n)$, thereby proving Kaneko's conjecture for finite multiple zeta values and its symmetric analogue.

Let $\X$ be a variety of groups. A group $G$ is $\CSX$ if its maximal $\X$-subgroups are malnormal. We prove that the amalgamated free product of two $\CSX$-groups with a malnormal amalgam and without any involutions is again $\CSX$. Using this, we describe conditions under which the HNN-extension of a $\CSX$-group is $\CSX$.

Meeting hard task-completion deadlines for many simultaneously offloading users is the binding challenge in mobile-edge computing (MEC). As the user count grows, the transmit power required to satisfy every deadline within per-user budgets rises until no feasible allocation exists. We take a feasibility-first view of this $K$-user problem in an uplink non-orthogonal multiple access (NOMA) MEC system served by a pinching-antenna system (PASS), whose radiating points establish strong line-of-sight (LoS) links that lower the received-power requirement. We formulate a latency-constrained joint optimization of transmit powers, CPU frequencies, and antenna configuration. We then derive a closed-form least-power feasibility test: the componentwise-least allocation meeting every deadline. It certifies feasibility with the largest attainable margin to the power budgets and defines a practical quality-of-service (QoS)-feasibility outage metric, the probability that no allocation meets all deadlines within budget. We solve the problem with a three-block alternating-optimization algorithm that is provably monotone-convergent and has polynomial complexity. Numerical results show that the proposed scheme sustains deadline feasibility at user loads where equal-power, orthogonal multiple access (OMA), and single-antenna baselines fail or degrade sharply, each by a mechanism predicted directly from the system model, whereas the proposed design degrades gracefully. Stress tests further show that feasibility is sustained under order-of-magnitude budget reductions, and that the pinching-antenna array itself enables constrained-budget operation. The result is a scalable, closed-form-initialized design for deadline-feasible offloading.

Sampling Gaussian measures restricted to nonlinear manifolds in high dimensions requires a proposal kernel that respects both geometric constraints and dimensional scaling requirements. We introduce tangent preconditioned Crank-Nicolson (pCN), which combines a manifold projection with a cancellation of the ambient Gaussian contribution, leaving only a geometric residual. Considering a shifted sphere with $\sqrt d$ scaling of radius and centre as a tractable benchmark embedded manifold, we prove that uncorrected tangent pCN results in a proposal noise scale of order $d^{-1/6}$ (compared to $d^{-1/2}$ for a tangent random-walk). We establish asymptotic optimal tuning for the geodesic expected squared jump distance which is maximised at an acceptance probability of 0.574 for the family of drift-corrected tangent proposals containing tangent pCN and tangent MALA. We also prove a joint diffusion limit for the parallel and orthogonal coordinates to the sphere's centre vector of tangent pCN. The orthogonal diffusion speed matches the limiting geodesic expected squared jump distance and is maximised at acceptance probability 0.574, whereas the parallel speed is maximised at 0.712, demonstrating that optimal tuning depends on the observable. We then identify an intrinsic precision correction characterised by potential Hessian and Ricci curvature that yields a proposal kernel with dimension-independent fixed step size limit. We discuss a generalisation to arbitrary hypersurfaces and present nondegenerate scaling of tangent pCN applied to conditioned diffusions under resolution refinement.

In this paper, we study the following Schrödinger-Newton system with a vanishing magnetic potential and an electric potential \begin{equation}\label{0.1} \left\{ \renewcommand{\arraystretch}{1.25} \begin{array}{ll} (-i\nabla +A(x))^2u+V(x)u=\Phi u, \ & x\in\mathbb{R}^3,\\ -\Delta \Phi =|u|^2, \ & x\in\mathbb{R}^3, \end{array} \right. \end{equation} where $i$ is the imaginary unit, $A:\mathbb{R}^3\to \mathbb{R}^3$ is a continuous magnetic potential, and $V:\mathbb{R}^3\to \mathbb{R}$ is a continuous electric potential. First, we show that the ground state energy of system with $A(x)\equiv0$ and $V(x)\equiv V_{\infty}>0$ is isolated. Using this fact together with a global compactness lemma, a barycenter function, and Brouwer degree theory, we prove that under suitable conditions on $A$ and $V$, system admits either a ground state solution or a high energy solution. Interestingly, whether a ground state solution exists is significantly affected by the magnetic potential $A$ and the electric potential $V$. Furthermore, we provide sufficient conditions that guarantee the existence of a ground state solution, as well as conditions under which no ground state solution exists and a high energy solution appears instead.

Rayleigh-Bénard convection (RBC) is a model for turbulent flow and challenging to simulate by direct numerical simulation (DNS). Adequate resolution of all relevant length scales requires very fine grids in space and stability restrictions can dictate small step sizes. In this paper, we investigate the use of parallel spectral deferred correction (SDC) as a high-order time-stepping method to reduce wall-clock times of direct numerical simulation of RBC. Performance results on JUSUF (CPUs) and JUPITER (GPUs) with two different codes show parallel SDC can offer some concurrency along the temporal axis plus larger maximum stable time steps. Taken together, this allows for a reduction of wall-clock time required for simulations over one free-fall time by having to compute fewer steps and being able to mitigate the higher cost per time step through temporal parallelization.

We study the one-dimensional nonlinear Schrödinger system \[ -u_i''-2\Big(\sum_{k=1}^N u_k^2\Big)^{p-1}u_i=\mu_i u_i \quad \mbox{in } \ \mathbb R,\ \ i=1,2,\cdots,N, \tag{S} \] where $u=(u_1,\cdots,u_N)\in H^1(\mathbb{R})^N$, $\mu_1\leq\mu_2\leq\cdots\leq\mu_N<0$, $1<p<3$ and $N\geq 2$ are arbitrary. For the cubic case where $p=2$ and $N\ge 2$, we establish the following structural results of the system $(S)$: (i) all nontrivial solutions can be classified completely; (ii) the linearized operator at any nontrivial solution is non-degenerate; (iii) exact $L^2$-mass formulas of all nontrivial solutions are derived in terms of distinct Lagrange multipliers $\mu_i$. As the applications of the above structural results, we first show that the system $(S)$ with $p=2$ does not have any orthonormal solution for any $N\ge2$. We then investigate a fermionic variational problem $J_N(p)$ with $N\ge2$, whose minimizers correspond to ground states of the system $(S)$ under orthonormal constraints. More precisely, we prove that there exists a sufficiently small $\delta_N>0$ such that $J_N(p)$ is attained for any $p\in(2-\delta_N,2)$, whereas $J_N(p)$ is not attained for any $p\in[2,2+\delta_N)$. These settle some conjectures of [Gontier–Lewin–Nazar, ARMA, 2021] and [Frank–Gontier–Lewin, CMP, 2021].

We consider multiple saddle-point linear systems with block-arrow structure. We analyze exact and inexact Schur complement based block diagonal preconditioner, by characterizing the eigenvalues of the preconditioned matrices in terms of the zeros of specific rational functions. Numerical results onto large tests coming from liquid crystal modeling and PDE-constrained optimization validate the obtained eigenvalue bounds

We construct a stationary weakly harmonic map from a domain in \R^5 into a compact Kähler surface for which a tangent map is weakly harmonic but nonstationary. The target manifold is the blow-up of \CP^1\times\CP^1 at four points lying in one fiber and arranged at the vertices of a regular tetrahedron. The nonstationarity of the tangent map arises from the break down of the tetrahedral symmetry in the defect measure concentrated on the vertices.

Maximum-likelihood decoding of linear codes over erasure channels can be considered as solving a linear system of a column submatrix of the parity-check matrix. In practice, Gaussian elimination is usually applied, and its computational cost depends on the structure of the matrix, especially the sparsity and the fill-in generated during the elimination. The chordal graph, or triangulated graph, first introduced by C. Berge, has a so-called perfect elimination ordering, indicating that if the primal graph of a matrix is a chordal graph, then there exists a perfect elimination ordering such that a special class of fill-in under Gaussian elimination following this ordering will be avoided. In this paper, we introduce chordal codes, i.e., codes equipped with parity-check matrices such that the primal graphs of the parity-check matrices are chordal. We prove that, for a chordal code, the decoding submatrix associated with any erasure pattern retains a chordal primal graph. So, the complexity of chordal elimination (Gaussian elimination following the perfect elimination ordering applied to the chordal code decoding) will be smaller than decoding a random code by Gaussian elimination. Also, we provide an upper bound on the complexity of chordal elimination.