Papers

Topology optimization seeks structures that minimize an objective function subject to constraints. While extensive research has focused on single-objective formulations, design is usually a trade-off between conflicting criteria. This naturally leads topology optimization to the multiobjective optimization field. Unfortunately, despite their long coexistence, interaction between these fields has remained limited. This work aims to bridge this gap. First, we provide a theoretical comparison of two common scalarization techniques, the weighted-sum and $\varepsilon$-constraint method, to the established Pascoletti-Serafini scalarization. Then, we perform extensive bi-objective numerical experiments, including minimization of volume, compliance, maximum stress and dynamic compliance. For each experiment, the four scalarization methods are aggregated, often alongside design space sampling and continuation, to better resolve the Pareto front. The major contribution of this work is the observation that the Pareto front consists of segments belonging to local fronts. Contrary to the smooth, convex fronts often reported in the literature, we show that this property can lead to discontinuous and nonconvex fronts. Finally, we numerically compare the approximation quality of the scalarization methods. The numerical examples demonstrate the robustness of the Pascoletti-Serafini scalarization against fronts with pronounced discontinuities, nonconvexities and high-curvature regions. In contrast, the weighted-sum and $\varepsilon$-constraint methods can produce clustering and gaps, depending on the shape of the front.

The Riviera model mimics a densifying settlement along the coastline. In the lattice version, houses are built sequentially in empty sites with the constraint that every newly built house has at least one empty neighboring site. The distribution of clusters of adjacent houses does not obey a closed set of evolutionary equations, but the void-cluster-void distribution does. We compute the latter and extract the cluster distribution from it. In the jammed state, when all voids have length one, the cluster distribution takes a simple closed form and exhibits factorial decay with cluster size. For finite systems, we use a static approach that directly analyzes jammed states. If the coastline is a finite segment, we determine the statistics of the number of empty sites in the jammed state (the average, variance, and higher cumulants). We also study a continuum version in which houses are built along the line, subject to a minimum separation from at least one neighboring house.

Motivated by a self-referential puzzle, we study sequences of voucher price tags in which each choice multiplies the cost of the following one. We connect the puzzle setting to classical permutation statistics, introducing the voucher cost alongside the related pairwise and loop costs. This perspective allows us to translate questions about budgeting into extremal problems on permutations. We review known results for permutations of ${1,2,\dots,n}$ and extend them to arbitrary sets of distinct non-negative price tags.

We construct diffeological moduli stacks $\mathscr{M}_{Dol}(X)$ and $\mathscr{M}_{dR}(X)$ parametrizing smooth families of Higgs bundles and those of flat bundles, respectively, on a compact Kähler manifold $X$. We then establish an equivalence of stacks between diffeological substacks $\mathscr{M}^\mathscr{H}_{Dol} \subset \mathscr{M}_{Dol}(X)$ and $\mathscr{M}^\mathscr{H}_{dR}(X) \subset \mathscr{M}_{dR}(X)$, whose fibres over the point are the categories of semistable Higgs bundles, with the usual condition on Chern classes, and of flat bundles on $X$, respectively. $\mathscr{M}^\mathscr{H}_{Dol}(X)$ contains families of semistable Higgs bundles to points of which the classical correspondence of coarse moduli spaces does not extend continuously. This shows that the equivalence we provide is, in a sense, a common extension, in the context of diffeological moduli stacks, of both the homeomorphism of coarse moduli spaces, and of the equivalence of categories between semistable Higgs bundles and arbitrary flat bundles.

In $\mathbb{R}^{3}$, we prove that $L^q\to L^p$ restriction estimates associated with smooth surfaces with negative Gaussian curvature hold for all $p>\frac{22}{7}$ and $q'<\frac{p}{2}$.

When a person's records appear in k independent data silos, each protected by (epsilon, delta)-differential privacy, standard composition yields a valid (k*epsilon, k*delta)-DP guarantee for the joint output. This worst-case bound, however, does not answer the concrete inference question: at what k can an adversary actually identify a target person? This paper develops the information-theoretic framework needed to answer that question. We introduce cross-silo person-level DP (XSP-DP), a Pufferfish-style privacy notion whose adjacency relation captures all records of a single person across all silos simultaneously, and verify that the standard basic composition bound carries over to this adjacency model. Within this framework we prove that de-anonymization undergoes a phase transition at k* = Theta(log n / epsilon^2) (population size n, per-silo RR parameter epsilon): a Fano lower bound shows any estimator fails for k << k*, while a matching maximum-likelihood upper bound shows the attack succeeds for k >> k*. An explicit XOR + randomized-response construction demonstrates information synergy: each silo's output is individually uninformative about the target, yet the joint mutual information is strictly positive. For non-coordinated binary randomized-response mechanisms, we prove that de-anonymization is inevitable once k exceeds the threshold, establishing that cross-silo coordination is necessary. These results provide a baseline threat model and Theta-level threshold for cross-silo inference attacks under local DP.

We establish quantitative stability of the De Philippis–Rindler rigidity on conical convex regions. By constructing $\mathcal{A}$-quasiconvex functions with a tailored concavity—which additionally yields a new, elementary proof of the original theorem for constant-rank operators—we prove a spreading inequality that strictly limits the angular concentration of $\mathcal{A}$-free measures. This geometric bound establishes that singular concentrations cannot cluster arbitrarily close to a direction outside the Tartar wave cone. As a consequence, we obtain a new $L^1$ compensated compactness result: $\mathcal{A}$-free sequences with uniformly bounded mass are forced to be equi-integrable, thereby preventing the formation of mass concentrations, provided their targets are asymptotically restricted to cones whose aperture is controlled by a power of the distance to the wave cone of $\mathcal A$.

We characterize the boundedness and compactness of the embedding $$ \operatorname{id}:\mathcal B \longrightarrow L^p(\mu),\qquad 0<p<\infty, $$ where $\mathcal B$ denotes the Bloch space and $\mu$ is a positive Borel measure on the unit disc. The criteria depend only on the dyadic Carleson tents. These results refine our earlier $L^2(\mu)$ characterization based on a full dyadic resolution of the disc and extend it to all positive exponents. The proof combines discretization on Carleson tents, representations of Bloch functions by series of analytic polynomials, and weighted matrix estimates. We also obtain corresponding boundedness and compactness characterizations for $$ \textrm{id}:\mathcal Q_p\longrightarrow L^q(\mu),\qquad 0<p\le1,\quad 0<q<\infty. $$ These results resolve the general $p$–Bloch–Carleson measure problem studied by Girela, Peláez, Pérez-González and Rättyä in 2008.

In this paper, we present a variational formulation of local configurational constraints that couple multiple beams with arbitrarily shaped cross-sections. Since this formulation requires no explicit interface to rotational degrees-of-freedom, it applies to any beam kinematics and finite element discretization. Here, we define the offset coordinates in a moving frame to constrain or release the relative position between connected beams. The present method is based on a first-order approximation of the Lagrange multiplier field in the cross-section, which limits the transferability of the joint to the resultant force and moment only. The multipliers are eliminated using a discrete null space method, which provides size reduction and improved conditioning of the system matrix. Further, we apply the developed formulation to a beam element based on extensible directors and to a brick element in nonlinear elastostatics. Several numerical examples are presented.

Universal coders process individual sequences without assuming that the source distribution is known. In this setting, uniformly generated sequences represent the most difficult test case: the source simulates pure randomness, contains no exploitable bias, and forces a frequency-estimating universal coder to infer the empirical composition entirely from the sequence itself. This paper reports that a Set Shaping Theory (SST) transformation systematically reduces the average universal coding length of uniformly generated sequences below the Krichevsky-Trofimov baseline N H_0(s) + R_KT(s). The transformation maps each input sequence s in A^N into an expanded sequence f(s) in A^(N+1), while storing the transformation index in the additional symbol in order to preserve reversibility. The comparison evaluates the exact Krichevsky-Trofimov baseline N H_0(s) + R_KT(s) against the shaped score (N+1)H_0(f(s)) + R_KT(f(s)), where H_0 is computed from the empirical frequencies of the individual sequence. A single unified transformation also yields reductions across distinct compression architectures, including adaptive arithmetic coding, enumerative coding, LZ78, adaptive Huffman coding, and adaptive ANS. These results support the interpretation of SST as a representation-level preprocessing layer that can structurally improve existing universal coders without requiring internal modifications to their coding mechanisms. All results reported in the article can be reproduced with the simulator available at https://sst-simulator.github.io/Set-Shaping-Theory-Simulator/.

Recently, linked partition ideals have attracted renewed attention. In this paper, we embed the gap-frequency partitions in which each part appears at most twice or three times into the framework of span one linked partition ideals, and derive refined generating functions for both cases. Furthermore, for gap-frequency partitions in which every part appears at most k times, we provide a purely combinatorial proof of the corresponding refined generating function.

We disprove the near-quadratic Elekes–Rónyai expander conjecture over $\mathbb R$. The counterexample is a fixed nonspecial quadratic polynomial, together with arbitrarily large finite sets of real algebraic integers on which its image has a fixed power saving from quadratic size. The main result of this paper is obtained by generative AI, particularly ChatGPT 5.5 Pro and the Rethlas system. The proof relies on a recent construction by OpenAI of an infinite tower of number fields.

Engineers often measure many quantities-speed, pressure, temperature, length-expressed in different physical units. The Buckingham Pi-grec theorem states that these variables can always be combined into a smaller set of dimensionless numbers whose values fully determine the system's behaviour. Identifying the appropriate dimensionless groups has traditionally required expert knowledge and physical insight. This paper shows that they can instead be discovered automatically from data, without prior knowledge of the governing physics. The key observation is that, after logarithmic transformation, measurements collected under different scalings of the same system lie on a low-dimensional manifold whose geometry is determined by the underlying dimensionless groups. Singular value decomposition (SVD) identifies this manifold directly from data. A subsequent search over integer-exponent combinations recovers candidate dimensionless quantities, while a repeating-variable filter retains only those constructed from the machine's characteristic scales. This procedure recovers familiar engineering groups, including the flow coefficient, head coefficient, and Mach number, while excluding equivalent but less interpretable alternatives. The method is demonstrated on a synthetic compressor dataset containing 16,000 measurements. Starting from raw dimensional variables and no physics input, it recovers the correct dimensionless groups to numerical precision and reproduces the compressor performance map with an error below 0.01%. More broadly, the work reveals a close connection between classical dimensional analysis and modern data-driven learning. Both rely on the same underlying algebraic structure, suggesting new approaches for building physical models that are simultaneously interpretable, scalable, and data-efficient.

Let (R,m) be a Noetherian local ring, I an ideal of R and M a finitely generated R-module. In this paper, we define and study E-depth and Ef-depth of M in I. We prove that E-depth (resp. Ef-depth under mild conditions) of M in I is the common length of all maximal sequential sequences (resp. maximal sequential f-sequences) of M in I. We show that E-depth and Ef-depth do not decrease under small pertubations. We describe the non sequentially Cohen-Macaulay locus nSCM(M) of M in terms of support and non Cohen-Macaulay locus of deficiency modules of M. Finally, we show that the dimension of non sequentially Cohen-Macaulay locus with respect to a sequential f-sequence does not increase under small pertubations.

While there is an extensive body of work characterizing the sample complexity of discounted cumulative-reward MDPs, finite sample analyses for average-reward MDPs have been limited, and most existing works rely on restrictive assumptions such as ergodicity or access to a generative model. In this work, we establish the first finite sample complexity guarantees from a single trajectory for weakly communicating average-reward MDPs. To this end, we study the dynamics of a single trajectory in weakly communicating MDPs and based on this analysis, we develop novel model-free methods. Notably, our value-based and policy-based methods provide finite sample complexity guarantees of $\widetilde{O}(1/\varepsilon^2)$ and $\widetilde{O}(1/\varepsilon^4)$ from a single trajectory in weakly communicating MDPs, respectively. Furthermore, we introduce the first model-free method that requires no prior knowledge of problem-dependent quantities for communicating MDPs.

We study a bootstrap percolation process on $\mathbb{Z}^d$ where each newly occupied point completes a copy of one of the patterns in some fixed collection $\mathcal{A}$. We aim, given $\mathcal{A}$, to find the smallest size of a percolating seed. We show that, for two patterns in two dimensions, in essence the answer is given by the mixed volume of the patterns' convex hulls. Our motivation for studying this setup comes from the theory of partial difference equations. Conversely, our considerations yield one consistency criterion for systems of two partial difference equations in two dimensions.

In this paper, the reconstruction of a linear differential operator of arbitrary order $n \ge 2$ is studied by using two types of spectral characteristics: (i) eigenvalues and weight numbers, (ii) $(2n-2)$ spectra. We prove the unconditional uniform stability of these inverse problems, generalizing the results of Savchuk and Shkalikov [Funct. Anal. Appl. 44 (2010), no. 4, 270–285] to $n > 2$. Furthermore, we for the first time obtain sufficient conditions of solvability for the higher-order inverse problem by $(2n-2)$ spectra. By applying our main results, we get new theorems on the necessary and sufficient conditions of solvability and on the uniform stability of the inverse problems for $n = 3$ and $n = 4$. Our approach is based on the method of spectral mappings, which provides a constructive solution of the inverse problems.

We study the problem of detecting a faint geometric signal hidden in an otherwise random graph. Formally, we consider a hypothesis testing problem in which, under the null, the observed graph is an Erdős–Rényi random graph $\mathcal{G}(n,q)$, while under the alternative a random geometric graph $\mathcal{G}(k,q,d)$ is planted on $k\le n$ vertices. The planted subgraph is generated from independent random points on the unit sphere $\mathbb{S}^{d-1}$, with edges determined by latent geometric proximity and calibrated to have edge density $q$. Our goal is to characterize the statistical and computational limits of detecting this hidden geometry. We derive sharp information-theoretic lower bounds that identify regimes where detection is impossible and provide algorithms that achieve these limits whenever detection is feasible. We further investigate the computational complexity of the problem and determine when efficient polynomial-time tests exist. The model exhibits an easy–hard–impossible phase transition: some regimes allow efficient detection, others permit detection only with computationally intractable procedures, and still others render detection impossible even with unlimited computational power. As evidence for the computational barrier, we prove that all low-degree polynomial algorithms fail throughout the conjecturally hard regime, demonstrating a sharp gap between statistical and computational feasibility.

Motivated by the mirror symmetry for circle compactified 4d $N=2$ theories, we propose a geometric framework for studying the representation theory of non-admissible $W$-algebras $W^{k}(\mathfrak g,f)$ at levels $k=-h^\vee+\frac{1}{n}\frac{m}{u}$, using the geometry of generalized affine Springer fibers $Sp_{\nu}(\tilde{g},f)$ with slope $\nu=u/m$. The central proposal is that each non-empty $C^*$-fixed locus, labeled by a double coset $\tilde w\in W_\nu\backslash\tilde{W}/W_f$, gives rise to simple modules whose highest weight is determined by the map $\tilde w\mapsto\tilde w(k\Lambda_0+\tilde\rho)-\tilde\rho$, while the dimension of the fixed variety encodes additional non-semisimple structure (logarithmic modules). We verify this correspondence in numerous examples, including $D_4$, $E_6$, $E_8$, and twisted theories of type $^3D_4$ and $^2A_3$, where our geometric counting reproduces known results for both admissible and non-admissible $W$-algebras.

Under cone containment and the central ray hypothesis, we prove a determinant majorization result for hyperbolic polynomials on Euclidean Jordan algebras. In the case of real symmetric $n \times n$ matrices, this recovers the main theorem of Harvey and Lawson [Duke Math. J. 174 (2025), no. 13, 2749–2763] and also yields a new $\sigma_2$ majorization result. Moreover, for every $3 \leq k \leq n-1$, we construct explicit hyperbolic polynomials satisfying cone containment and the central ray hypothesis but for which the analogous $\sigma_k$ majorization fails.

We prove a quantitative version of Oppenheim's conjecture in the function field setting. In order to do so, we compute the higher moments of the Siegel transform. In particular, we find an optimal bound on the variance of the number of lattice points in a set. Moreover, we compute the exact variance of the number of lattice points in a ball, which is of independent interest.

In this article we present new and improved results for the ratio between the independent domination number and the domination number in graphs with bounded degree. We present a general formula, that, for a fixed maximum degree, allows to compute an upper bound for this ratio as a function of an upper bound $\beta |V|$ for the independent domination number. We also apply this formula to several known upper bounds for the independent domination number. Furthermore we present constructions giving lower bounds for the best possible upper bound in various classes of graphs with bounded degree.

We propose a structure-preserving parametric finite element method for the constrained Helfrich flow of closed curves and surfaces. The proposed method is based on a two-stage velocity-splitting strategy. In the first stage, the normal velocity is computed from a curvature evolution equation, with the volume and surface area constraints imposed softly in terms of the normal velocity. This step approximates the gradient-flow structure of the Helfrich flow, and its fully discrete parametric finite element approximation leads to a linear system and yields an unconditional energy dissipation estimate at the fully discrete level. In the second stage, the surface mesh is updated by combining the computed normal velocity with a BGN-type tangential velocity. A time-weighted interface normal and an area-correction multiplier are also used to enforce exact preservation of the enclosed volume and surface area. This correction step leads to a nonlinear system, which can be efficiently solved by Newton iteration. The resulting method simultaneously achieves energy decay, exact geometric conservation, and good mesh quality. Numerical experiments for two-dimensional curves and three-dimensional surfaces, including nonsmooth initial data and nonzero spontaneous curvature, are presented to demonstrate the accuracy, robustness, and structure-preserving properties of the proposed method.

We prove the exact octonary covering-code value $K_8(4, 2) = 23$ in Lean 4. The upper bound is given by an explicit 23-word radius-two code in $(Fin\:8)^4$ , checked over all $8^4$ ambient words. The lower bound excludes covers with at most 22 words. A fiber-counting and missing-pair argument first rules out covers with at most 21 words. In the remaining 22-word case, the proof reduces a hypothetical cover to six missing-pair graphs coming from the coordinate-pair projections. Fiber-counting arguments constrain these graphs, and two Lean-checked Linear RAT (LRAT) refutations of stored conjunctive-normal-form (CNF) instances force a common 3 + 3 + 2 block structure. This structure is incompatible with a 22-word cover: the two three-symbol components already force 18 codewords, while the remaining two-symbol component would require a binary strength-two array of length four with at most four rows, which is impossible. The result is packaged as a proof-carrying Lean artifact: the explicit upper bound, structural lower bound, CNF instances, and LRAT refutations are checked inside Lean, with no external SAT solver used during proof replay.

It is now well known that the classical notion of admissible weak solutions (also known as weak entropy solutions) does not restore uniqueness for the multi-dimensional compressible Euler equations. Indeed, convex integration has shown that admissible weak solutions are in general highly non-unique. This has motivated additional selection criteria intended to rule out the counterintuitive solutions generated by convex integration. In this paper we prove that the local least action criterion introduced by H.~Gimperlein, M.~Grinfeld, R.~J.~Knops and M.~Slemrod does not serve as a proper selection criterion, since it fails to select the solution that is intuitively expected to be physically relevant.

The Riemann–Roch theorem for graphs, due to Baker and Norine, is a foundational result establishing a powerful analogy between finite graphs and algebraic curves. We describe a complete formal proof of this theorem implemented in the Lean 4 theorem prover. Our formalization includes the existence and uniqueness of q-reduced divisors, a modified form of Dhar's burning algorithm, the bijection between acyclic orientations with unique source and maximal superstable configurations, and Clifford's theorem. We also include several challenges for future formalization.

This paper investigates the impact of quadrature accuracy for volume and face integrals in the finite element method using $p$-th order polynomial shape functions for elliptic problems with mixed Dirichlet and Robin boundary conditions. The optimal $p$-th order $H^1$-convergence is maintained when numerical integration with algebraic precision at least $2p-2$ for volume terms and at least $2p-1$ for face terms is adopted. For $L^2$-error, we achieve optimal $O(h^{p+1})$ convergence when using quadrature rules of precision no less than $\max\{p,2p-2\}$ for volume terms and no less than $2p-1$ for face terms. Of particular significance, we present two examples to show that the above result on $L^2$-error is sharp for the linear FEM ($p=1$). When reduced to the case of Dirichlet boundary condition, our results yield improved dependence on the given data compared to the classical results established by Ciarlet, et al. Numerical experiments are provided to illustrate the theoretical findings and confirm the necessity of specified quadrature accuracy and data regularity.

We study finite-horizon one-gradient realizations with the Chebyshev minimax terminal residual on $[\mu,L]$. Under time-dependent Hessian perturbations, the terminal first variation is governed by a time-ordered spectral kernel $K_s(\lambda,\nu)$; its sharp $\ell_2$ gain is $A_N$. For the prefix-exact Chebyshev recurrence, $$A_N^{pref}=\frac4{\sqrt3}\frac{N^{3/2}}{L-\mu}\epsilon_N^\star(1+o(1)),$$ and this is sharp in the causal two-term class with Chebyshev exactness at every prefix. For terminal-only exactness, Jacobi coordinates give $P_N=2^{1-N}T_N$: the spectrum is fixed at the midpoint Chebyshev nodes, while the spectral weights parametrize the realizations. The sine weights give a final-exact Jacobi method with the same terminal residual and $$A_N(J_N^{\sin})=2\sqrt{c_{\sin}}\frac{N^{3/2}}{L-\mu}\epsilon_N^\star(1+o(1)), 2\sqrt{c_{\sin}}\approx2.1379<4/\sqrt3.$$ Thus the Chebyshev terminal polynomial does not determine the first-order Hessian-drift gain. Momentum is necessary for this order: every Chebyshev-exact stepsize schedule has $A_N\ge2\sqrt{2/3} N^2\epsilon_N^\star/(L-\mu)$. Beyond first order, the response to drift $E_s=H_s-H$ of size $\delta=\max_s\|E_s\|$ is a finite polynomial expansion, and for both realizations the terminal error is at most $\epsilon_N^\star T_N(1+2\delta/(L-\mu))$, the Chebyshev bound of the interval widened by $\delta$. This bound is attained, holds for objectives with Lipschitz Hessian, and shows that drift reduces the accelerated rate exponent by the factor $1-\sqrt{\delta/\mu}$. The two realizations thus share the worst-case response and differ in the $\ell_2$-in-time response measured by $A_N$. The experiments show the finite-horizon effect: lower stochastic curvature overhead, larger admissible-block frontiers, accurate time-varying quadratic predictions, and lower restart cost on an endpoint-coupled smooth strongly convex GLM.

Covering problems in coding theory are closely related to finite geometry through the interpretation of the columns of parity-check matrices as point sets in finite vector spaces. Motivated by the recent notion of generalized covering radii of linear codes introduced by Elimelech, Firer and Schwartz, we develop a geometric framework for these parameters. We introduce $(\rho,t)$-saturating sets and show that they are precisely the finite-geometric counterparts of linear codes whose $t$-th generalized covering radius is at most $\rho$. We study the structure of these sets and show that the extremal case $\rho=t$ coincides with the notion of $t$-strong blocking sets. Thus, $(\rho,t)$-saturating sets interpolate between classical saturating sets and strong blocking sets. We provide several equivalent formulations, including affine and dual Grassmannian criteria, derive lower bounds on their size, and give constructions from strong blocking sets, graphs and projective configurations.

Let $\mathbb{F}$ be a field of characteristic different from $2$, and let $M_n(\mathbb{F})^+$ denote the Jordan algebra of all $n\times n$ matrices over $\mathbb{F}$ with product $X\circ Y:=(XY+YX)/2$. We prove a rigidity theorem for $M_n(\mathbb{F})^+$, $n\ge2$: if $\mathcal{J}$ is any $2$-torsion-free Jordan ring and $\phi:M_n(\mathbb{F})^+\to\mathcal{J}$ is a Jordan multiplicative (product-preserving) map, then $\phi(0)$ is an idempotent and $X\mapsto\phi(X)-\phi(0)$ is either zero or an injective Jordan ring homomorphism. Thus, up to an idempotent constant, preservation of the Jordan product alone forces additivity and the zero-or-injective dichotomy. When specialized to associative codomains, the theorem yields the Jacobson–Rickart decomposition into homomorphic and antihomomorphic parts. In particular, for maps $M_n(\mathbb F)^+\to M_k(\mathbb K)^+$, where $\mathbb K$ is a field of characteristic different from $2$, we also obtain a block normal form governed by finite-dimensional $\mathbb K$-representations of $\mathbb F$, together with a criterion for the existence of nonconstant maps.