Papers

We study the identification of effective coefficients for wave equations in heterogeneous media. Such equations arise in the modeling of spatio-temporal metamaterials, where the underlying material properties exhibit variations in both space and time. While homogenization provides effective models in the asymptotic regime of vanishing microscopic scales, determining macroscopic parameters from observations of wave propagation remains challenging. We introduce an optimization-based method that identifies a constant effective coefficient by minimizing a cost functional. The approach is designed for situations in which the underlying space-time-dependent coefficient is unknown, while the solution is available in space and time by measurements. We extend optimization-based coefficient identification techniques from elliptic multiscale problems to wave equations and prove that, provided a homogenized limit exists, the identified coefficient converges to the homogenized coefficient as the microscopic scale tends to zero. Furthermore, we establish convergence of the corresponding effective solution towards the heterogeneous solution, including strong convergence in $L^2$ and weak convergence of first-order space and time derivatives. Numerical experiments further demonstrate the performance of the method for non-periodic space-time heterogeneous media, including cases for which a homogenized limit is not known to exist.

We derive an explicit one-step kernel formula for the kinetic Fokker-Planck equation. The construction begins with a reference dynamics admitting an explicit transition kernel and uses a carefully chosen auxiliary weight to encode the drift perturbation. The resulting kernel formula provides an explicit approximation of the density evolution. In a weighted phase-space setting, we establish local weak consistency and first-order finite-time weak convergence under suitable regularity and moment assumptions. We also give a formal variational interpretation of the formula in terms of a regularized Wasserstein-type proximal operator and present analogous constructions for a broader class of parabolic equations. Numerical experiments illustrate the accuracy of the kernel approximation and its application to deterministic particle schemes for kinetic sampling.

In this work, we present two novel strategies to impose high-order boundary conditions on fixed and moving curved domains, approximated with piecewise affine triangulations. Achieving high-order accuracy on curved domains requires tackling both the PDE discretization error and the geometrical error simultaneously. While the former can be reduced by employing high-order numerical methods such as finite volume and discontinuous Galerkin, the latter demands either a high-order parametrization of the physical domain or a consistent approximation of the boundary conditions. Minimization-based approaches like the Reconstruction for Off-site Data (ROD) method allow one to skip the construction of high-order curvilinear meshes by defining high-order consistent boundary conditions on a computational boundary that does not match the physical one. The ROD approach mitigates the second-order geometrical error by retrieving a modified polynomial in each boundary cell, which enforces the boundary conditions exactly on the physical boundary. However, the standard ROD method requires the inversion of a local linear system, whose cost grows with the polynomial degree and mesh refinement. Inspired by a recent one-dimensional analysis, we show that ROD-type approaches can be recast as simple polynomial corrections, applicable without any linear system inversion. This greatly simplifies the development of minimization-based boundary treatments and reduces the associated computational cost. To prove the wide applicability of our strategy, we develop it within a Runge-Kutta discontinuous Galerkin framework and an ADER arbitrary-Lagrangian-Eulerian finite volume framework for compressible flows on fixed and moving domains. Several numerical experiments with Dirichlet and slip-wall boundary conditions are presented, with convergence analysis up to fifth order in both 2D and 3D.

We prove that a generic spherical parameter for the graded affine Hecke algebra with equal parameters is unitary if and only if its normalized intertwining forms are positive on the reflection representation and on the irreducible constituents of its second symmetric power. The proof combines a signature formula for the reflection representation, Yu's uniform two-wall operation, a simply-laced Jantzen recursion, and a root-poset analysis. In types B_n and C_n, we prove more sharply that the reflection representation V and the traceless-diagonal constituent of Sym^2(V) suffice. We also give a shorter second proof when arbitrary Weyl-group types are allowed. The consequence of the first proof (via Barbasch-Vogan petite K-types) is that the generic spherical unitary dual of a Chevalley group over the real numbers or a nonarchimedean local field, and for simply-laced groups and complex symplectic groups also over the complex numbers, is independent of the field. Moreover, the answer has a simple uniform description as the Weyl conjugates of a disjoint union of a power of 2 alcoves in the affine Weyl group arrangement in the fundamental Weyl chamber, where the exponent is the matching number of the Dynkin diagram of the subsystem of short coroots.

We construct noncommutative mirrors from the bulk-deformed Floer theory of compact immersed Lagrangians equipped with higher rank flat bundles, whose Floer theory can have non-zero curvature terms due to contributions of stable constant discs. For a nodal union of Lagrangian spheres, codimension-two bulk cycles meeting the compact core produce the additive deformed preprojective relations and the corresponding framed Maurer-Cartan spaces at nonzero complex moment-map levels. In particular, our localized mirror functors produce noncommutative ADHM and Nakajima monadic complexes. We obtain local charts and gluing maps of the noncommutative mirror varieties via explicit computations of Lagrangian deformation spaces and Fukaya isomorphisms between them. These constructions recover noncommutative spaces studied by Kapustin-Kuznetsov-Orlov, Baranovsky-Ginzburg-Kuznetsov, and Kawamata, with further examples in dimension three. We also establish the categorical framework in the unobstructed setting. For a compact $\mathbb Z$-graded unobstructed immersed Lagrangian $\mathbb{L}$ without nonpositive-degree immersed generators, Koszul duality identifies the derived Fukaya subcategory generated by $\mathbb{L}$ with the derived category of modules with finite-dimensional total cohomology over its extended localized mirror $\tilde{\mathcal{A}}_{\mathbb{L}}$. This construction starts from the Floer algebra of the compact reference brane and does not require a global wrapped generator. Applications to plumbings recover known split-generation results for compact objects.

Let $\{0,1\}^{n}$ denote the set of all $n$-dimensional vectors whose components are either $0$ or $1$. For any two nonempty subsets $\mathcal{X},\mathcal{Y}\subseteq\{0,1\}^{n}$, the pair $(\mathcal X,\mathcal Y)$ is called an $n$-dimensional unique-sum pair if each pair $(\boldsymbol x,\boldsymbol y)\in\mathcal X\times\mathcal Y$ can be uniquely determined from the arithmetic sum $\boldsymbol x+\boldsymbol y$. Let $M(n)$ denote the maximum value of $|\mathcal X||\mathcal Y|$ over all $n$-dimensional unique-sum pairs $(\mathcal X,\mathcal Y)$. In 1969, Lindström proved that $$ \frac{1}{2}(1+\log_2 3) \leq \lim_{n\to\infty}\frac{1}{n}\log_2 M(n) \leq 3/2. $$ Since then, the lower bound $(1+\log_2 3)/2$ has been successively improved, whereas the upper bound $3/2$ has remained unchanged. In this paper, we establish an explicit upper bound whose numerical value is approximately $1.4884$. To the best of our knowledge, this is the first strict improvement over Lindström's upper bound $3/2$. Towards this end, we develop a coordinate projection approach that constructs a lower-dimensional unique-sum system from a unique-sum pair. By combining the results obtained from this approach with a relaxed form of a necessary condition established by Ordentlich and Shayevitz on unique-sum systems, we establish the above improved upper bound.

In this note, we show that a weak mean curvature flow with critical forcing term obtained by Liu–Tonegawa (2024) satisfies the BV-type area change formula, that is, their flow is not only a Brakke flow but also a generalized BV flow, which is proposed by Stuvard–Tonegawa (2024). To establish this result, we identify minimal conditions under which a Brakke flow satisfies the area-change formula. As an application of our main theorem, we derive a lower bound for the extinction time of generalized BV flows with a critical forcing term. We also outline the proof of a compactness theorem for generalized BV flows.

In this paper we derive new Delsarte-type linear programming bounds for $q$-ary $(r,\delta)$-locally recoverable codes (LRCs) with three attributes: first, the variable set is comparable in size to that of the classical Delsarte LP; second, our LP exploits the higher-order information forced by the local-distance condition through order \(\delta-2\), in the sense that for nondegenerate linear codes, its balanced base part gives exactly the same dimension bound as the symmetrized refined-weight LP of Gruica, Jany, and Ravagnani, while the additional constraints, nonvacuous whenever $\delta \ge 3$, give a further strengthening; and third, it applies to general $(r,\delta)$-LRCs, linear and nonlinear alike. Extensive computations over binary and ternary alphabets show that the convex-hull LP yields improvements not captured by the previous LP and often sharpens the shortening and generalized Singleton bounds.

Quantum cryptographic primitives beyond key distribution remain a less well understood area of research. In classical cryptography, one-way functions underpin nearly all standard cryptographic protocols, motivating the search for meaningful quantum analogues and for a clear understanding of the physical and computational mechanisms that could enforce one-wayness. In this article, we review quantum one-way functions and a range of closely related quantum-state primitives, including one-way state generators, pseudorandom quantum states, and efficiently indistinguishable pairs of states. We discuss both computational and information-theoretic notions of quantum one-wayness, emphasizing the different adversarial models and security assumptions that underlie these constructions. We compare and contrast the various proposed primitives, and clarify their conceptual relationships. Particular emphasis is placed on questions of physical realizability, experimental feasibility, and robustness to noise. Finally, we outline open problems and future directions toward the development of practical quantum cryptographic primitives beyond key distribution, and the emergence of a broader quantum-cryptographic ecosystem.

Let $f_{\ell}(n)$ be the largest size of a family $\mathcal{A}\subseteq2^{[n]}$ such that no member has size divisible by $\ell$, while the intersection of every two distinct members has size divisible by $\ell$, and let $\omega(\ell)$ denote the number of distinct prime divisors of $\ell$. For any prime power $\ell$, the classical answer is $f_{\ell}(n)=n$. When $\omega(\ell)\geq 2$, Bukh, Chao, and Zheng recently proved $\omega(\ell)n-O_{\ell}\left(n^{\frac{\omega(\ell)-2}{\omega(\ell)-1}}(\log n)^{C_{\ell}}\right)\leq f_{\ell}(n)\leq\omega(\ell)n-2\omega(\ell)\log n+11$ for some $C_{\ell}>0$. When $\ell$ has at least two distinct odd prime divisors, they further used Fourier analysis to improve the upper bound to $f_{\ell}(n)\leq\omega(\ell)n-(2\omega(\ell)+\varepsilon_{\ell})\log n$ for some $\varepsilon_{\ell}>0$, provided that $n$ is sufficiently large in terms of $\ell$ . For every fixed $\ell$ with $\omega(\ell)\geq2$, we prove \[ f_{\ell}(n)\leq\omega(\ell)n-\Omega_{\ell}(\log n\log\log n) \] for large $n$. The upper bound relies on a submatrix lemma of Bhowmick, Dvir, and Lovett, which is based on the bounded-torsion polynomial Freiman–Ruzsa conjecture recently proved by Gowers, Green, Manners, and Tao.

We solve cyclic finite-dimensional quantum histories for arbitrary time-dependent unitary steps, without assuming that one step has finite order. The propagation Hamiltonian is a unitary connection Laplacian on a cycle; its complete gauge invariant is the monodromy $M=U_{L-1}\cdots U_0$. Its spectrum is $\lambda_{a,k}=1-\cos((2\pi k-\theta_a)/L)$, where $e^{i\theta_a}\in\mathrm{spec}(M)$. Thus the exact history sector is isomorphic to $\mathrm{Fix}(M)$, while frustration, the gap above a nonempty zero-energy sector, the determinant, and the finite-temperature trace are obtained in closed form. Ordinary spectral data recover the multiset of monodromy phase cosines but not phase orientation; low energy certifies proximity to an exact relational history. We then define the predictive quotient of a sharp finite clock relative to an accessible operator system as the unique coarsest event alphabet preserving all conditional statistics on a history sector. A finite-error theorem shows that threshold clustering recovers this quotient when the minimum diamond separation of inequivalent event channels exceeds four times the estimation error, and proves an optimal record-count bound. With full matrix access and homogeneous step $U$, the minimal number of clock events is the projective order of $U$. We distinguish the normalizer of the clock algebra from transformations preserving the coherent history code and classify oriented exact sharp clock changes by $U(r)\times\mathbb{Z}_L$ on a rank-$r$ history sector; without orientation the cyclic factor becomes dihedral. Reversible changes of full-information clock fibers are necessarily unitary, so irreversible coarse-graining is not exact clock covariance. Minimal realizations of a complete history Gram kernel are uniquely unitarily equivalent, with a finite-data Procrustes bound. Independent finite-matrix code verifies the main results.

Let $\mathcal{P}$ be a three-dimensional convex polytope resting with one of its faces on the plane. At each step, $\mathcal{P}$ is allowed to roll over a randomly selected edge of the face currently lying on the plane, until the adjacent face comes to rest on the plane. The trace of $\mathcal{P}$ is the set of all points of the plane that can be reached by a vertex of $\mathcal{P}$, starting from a fixed initial position and performing a finite sequence of rolls. We prove that if the trace of $\mathcal{P}$ has a convergent subsequence, then, with probability one, the set of points reached by the vertices of a randomly rolling copy of $\mathcal{P}$ is everywhere dense in the plane. This settles a conjecture of Hegyvári.

We establish the existence of a positive solution for a one-dimensional Minkowski-curvature equation with mixed boundary conditions and a sign-changing nonlinearity. A nonnegative linear shift and the Green kernel of the associated linear problem produce an invariant cone. Principal-eigenvalue comparisons and the fixed point index give cone expansion near zero and compression at infinity. A globally defined auxiliary equation and a first-contact argument show that the resulting solution has slope below one and hence solves the original equation. We also derive a directly verifiable asymptotic criterion.

We prove that every $\mathcal{C}^{1,\alpha}$ self-map of a compact Riemannian manifold, in arbitrary dimension, admits a symbolic extension. This gives a positive answer to the conjecture of Downarowicz and Newhouse in arbitrary dimensions. Our method is based on analyzing the small singular values of the linearized orbit operator.

We consider a class of mixed anisotropic and nonlocal singular quasilinear elliptic problem associated with Muckenhoupt weights. The presence of the singular nonlinearity, which blows up near the origin along with its interaction with anisotropy, weighted degeneracy, and nonlocal diffusion creates significant analytical challenges. We employ monotone approximation, weighted Sobolev embeddings, compactness, and variational methods to establish the existence and uniqueness of weak solutions under suitable assumptions on the datum. Further, we characterize the best constant in an associated weighted mixed anisotropic and nonlocal Sobolev-type inequality, prove that it is attained, and show that the normalized weak solution is the unique extremal. These results are new even in the mixed weighted Laplace case \(p=2\).

Let $X$ be a topological space. For each transfinite cardinal number $\aleph_\alpha$, we define a topology $C_{\aleph_\alpha}(X)$ on the ring $C(X)$. With $\aleph_\alpha=\aleph_0$, $C_{\aleph_\alpha}(X)$ reduces to the space $C_p(X)$. We prove that for $\aleph_\alpha\geq \aleph_1$, $C_{\aleph_\alpha}(X)$ is pathwise connected if and only if it is connected if and only if $X$ is pseudocompact. Here, we define $\aleph_\alpha$-separable space and furthermore, we show that $X$ is $\aleph_\alpha$-separable when and only when $C_{\aleph_\alpha}(X)$ is metrizable when and only when it is a sequential space. Later we introduce two new cardinal functions, namely $cc_{\aleph_\alpha}(X)$ and $ac_{\aleph_\alpha}(X)$ which turned out to be the character and pseudocharacter of $C_{\aleph_\alpha}(X)$ respectively. At the end of this article we show that a number cardinal functions associated with the space $C_{\aleph_\alpha}(X)$ are equal.

We study the well-posedness of the boundary layer for a pseudo-plastic fluid by energy methods under Oleinik's monotonicity assumption. We need to address two main difficulties: derivative loss and the difficulty caused by viscosity. For the first difficulty, we borrow the cancellation mechanism proposed by Masmoudi and Wong [CPAM, 2015]. For the second difficulty, we combine the monotonicity assumption and Faà di Bruno formula to obtain precise control over the high-order derivatives of viscosity, which is the main contribution of this paper.

A linear computation broadcast (LCBC) problem comprises $K$ users (receivers) and a transmitter. The users wish to compute various (vector) linear functions of a common dataset, and possess in advance heterogeneous side information corresponding to various other linear computations. The goal for the broadcast transmitter, who knows the dataset and all desired and side-information functions, is to satisfy all demands as efficiently as possible. Prior work has explored the information-theoretic capacity of LCBC for zero-error finite-field computation over an ideal (noiseless) broadcast channel (BC). This paper develops a wireless LCBC (WLCBC) framework for the Gaussian MIMO BC with noisy receiver-side information under a transmit power constraint and arbitrary antenna configurations. In the WLCBC setting, a Gaussian source is linearly precoded for broadcast; each user requests a linear function of the source and forms an estimate by linearly combining its channel observation with its noisy side information. Assuming perfect channel state information, we cast the centralized joint linear transceiver design as a weighted sum-MSE minimization problem and propose an efficient alternating optimization algorithm. For a fixed precoder, the optimal decoders are the linear minimum mean-square error (LMMSE) estimators. For fixed decoders, the precoder update reduces to a convex quadratically constrained quadratic program which leads to a semi-closed-form solution parameterized by a single dual variable. The resulting algorithm guarantees a monotonic decrease in the objective and convergence of the weighted sum-MSE (WSMSE) objective sequence. Simulations demonstrate pronounced robustness gains over natural baselines obtained from prior works.

In this paper we first introduce and study the notions of pointwise convergence and uniform convergence of sequences of partial metric valued functions. Also introducing the notions of pointwise Cauchyness and uniform Cauchyness of sequences of partial metric valued functions, we study their relationship with pointwise convergence and uniform convergence. Further we introduce and study the notions of statistical pointwise convergence and statistical uniform convergence of sequences of partial metric valued functions including their interrelationship. Also, introducing the notion of equi-statistical convergence of sequences of partial metric valued functions we study its relationship with statistical pointwise convergence and statistical uniform convergence.

We characterize four groups of Clifford circuits for Calderbank–Shor–Steane (CSS) codes that are relevant to fault-tolerant logical operations. First, we show that every code-preserving Clifford circuit is a product of Z-diagonal circuits, composed of S and CZ gates, and their X-basis analogues. Second, we define the two-fold transversal group, generated by depth-one two-local code-preserving circuits, and show that each of its elements can be expressed as a product of layers consisting of either Z-diagonal, X-diagonal, or CNOT gates. As a corollary, every transversal gate is a product of three transversal diagonal circuits; for connected non-self-dual codes, two such circuits suffice. We further show that every code-preserving automorphism circuit, consisting of single-qubit Clifford gates and permutations, has a normal form comprising a Hadamard layer, a permutation, and two diagonal circuits. We also define a two-fold automorphism group, in which a depth-one two-local circuit may be code-preserving up to a permutation, and show that its logical image can be larger than that of the two-fold transversal group. For 231 CSS codes, we provide explicit generators and determine the logical image of the two-fold transversal group. We tabulate 173 codes whose full logical Clifford group is generated by two-fold-transversal circuits, including codes of distances 3, 4, 5, 6, 8, 10, 12, and 13, with respective best rates 2/5, 3/4, 34/77, 17/28, 14/25, 1/27, 3/56, and 1/33. We construct families of CSS codes from bipartite grids, cut-complements, and quadrics, and census the self-dual codes invariant under rank-3 permutation groups. Many of these codes realize the full logical Clifford group in this way.

Constrained density functional theory (CDFT) provides a powerful framework for describing electronically excited and charge-localized states, which underlie a broad range of physical and chemical phenomena. However, the discretized optimization problems arising from CDFT calculations remain challenging, owing to the presence of both the Stiefel manifold constraint and additional nonconvex quadratic constraints. Existing algorithms either fail to enforce the quadratic constraints with high accuracy or face convergence issues due to double-loop iterative structures. In this paper, we first derive a subspace-splitting reformulation that decouples the two groups of constraints, by exploiting the inherent rotation invariance and introducing a nonlinear subspace alignment constraint. Based on this reformulation, we propose a single-loop damped alternating direction method of multipliers, called DASSP. To the best of our knowledge, DASSP is the first algorithm for CDFT calculations with rigorous convergence guarantees. Each iteration of DASSP comprises a spectral minimization step, a projected gradient step, and a damped dual ascent step, all of which admit efficient implementations. Numerical results on synthetic and realistic CDFT problems demonstrate that DASSP attains high feasibility accuracy and exhibits favorable efficiency without compromising robustness. We expect that this work will pave the way toward reliable and efficient large-scale CDFT applications.

In this paper, we develop a unified Hamiltonian framework for dynamical systems with multiple independent dissipation channels based on q-contact geometry. As an application, we embed an elastoplastic damage model with two internal variables into the non-uniform 2-contact framework, explicitly construct the underlying geometric structure, derive the effective equations of motion, and validate the theoretical predictions through numerical simulations, which confirm the exponential dissipation law and the associated rescaled conserved quantity.

We consider linear regression models fitted by minimizing Bregman losses of the form \[ \frac{1}{n}\sum_{i=1}^n \left[ \phi(y_i)-\phi(x_i^\top\theta) -\phi'(x_i^\top\theta) \bigl(y_i-x_i^\top\theta\bigr) \right], \] where \(y_i\) is the observed response and \(x_i^\top\theta\) is the linear prediction. Even when the generating potential \(\phi\) is strongly convex, the resulting regression objective may be nonconvex in \(\theta\). This creates a gap between the convexity of the generating potential and the optimization geometry of the fitted model. The Hessian can be written as a weighted Gram matrix whose weights depend on the derivatives of the potential and the current residuals. This representation gives simple conditions for local strong convexity, smoothness, and conditional linear convergence of gradient descent. For the quadratic–quartic potential, we derive an exact scalar convexity condition, identify the interval of negative curvature, and obtain local and global sufficient conditions for positive curvature. Numerical experiments indicate that these scalar conditions may fail while the full Hessian remains positive definite at the evaluated points. They also indicate that the range of tested gradient-descent step sizes leading to convergence decreases as the quartic parameter grows. These results characterize how residual-dependent curvature interacts with the generating potential and the design matrix in reverse Bregman regression.

Consider the Gaussian Volterra process introduced by Mishura and Shklyar \cite{MS22a,MS22b}, $$ X(t) = \int_0^t r^\alpha \left( \int_r^t u^\beta (u-r)^\gamma\,du \right)dW_r, \qquad t\ge 0, $$ where $$ \alpha>-\frac12, \quad \gamma\in\left(-1,-\frac12\right), \quad H:=\alpha+\beta+\gamma+\frac32>0. $$ We obtain two-sided estimates for the small ball probabilities of $X$. As applications, we prove Chung's laws of the iterated logarithm (Chung's LILs) at every fixed point $t>0$, at the origin, and at infinity. The fixed-time result follows from the small ball estimates and the Lamperti transformation, whereas the results at the origin and infinity follow from Talagrand's lower-class criteria \cite{talagrand1996lower}. These results show that $\gamma+\frac32$ determines the local roughness and the small ball exponent, $\alpha+\beta$ determines the scale of local fluctuations at fixed positive times, and $H$ governs the self-similar scaling at the origin and infinity.

In this paper, we consider the periodic problem to steady viscous Burgers equation with an periodic and external force. Our main aim is to construct periodic solutions to this model which are uniformly bounded with respect to the viscosity via a direct Fourier series approach. Firstly, as an example, starting from a special solution of the non-viscous Burgers equation with a specific force, we solve the approximate solutions to the viscous Burgers equation in an explicit way. Secondly, we construct the solution to the periodic problem of the viscous Burgers equation with the external force by solving the initial value problem to a second ordinary differential equations, which is uniformly bounded with respect to the viscosity. Compared with Jauslin-Kreiss-Moser's result, our approach provides an explicit constructive procedure for the periodic solutions and yields a sharper convergence rate (up to higher order) when the viscosity vanishes.

We introduce Potential Matching Optimal Transport (PMOT), a potential-flow framework for general $p$-cost optimal transport with $c_p(x,y)=\|x-y\|^p$. PMOT parameterizes the CNF velocity field with a scalar potential in the generalized Benamou–Brenier form for the chosen exponent $p$. It trains the potential gradient with a self-induced matching loss along straight bridges determined by the model's own endpoints, while allowing flexible terminal distribution matching. Our main result establishes zero-loss exactness: under the stated regularity, exact terminal matching, and uniqueness assumptions, any zero-loss solution satisfies the generalized Benamou–Brenier optimality system and recovers the corresponding $p$-optimal transport map and dynamics. On synthetic benchmarks, PMOT learns $p$-specific maps that agree with the corresponding $p$-matched OT references. It also remains competitive as a likelihood-based density model on high-dimensional tabular data, and an MMD-based color transformation experiment demonstrates flexible sample-based terminal matching.

We prove that the Numerical Terao Conjecture holds for even-degree conic-line arrangements having only ADE singularities. We then show that the conjecture fails in the broader quasi-homogeneous setting once ordinary quadruple points are admitted, by constructing a degree-nine counterexample consisting of a pair of conic-line arrangements, each with seven lines and one smooth conic, with the same weak combinatorics \[ W(\mathcal{CL}) = (7,1;\,8A_1+D_4+4X_9). \] We show that one curve is free with exponents $(4,4)$, whereas the other is nearly free with exponents $(3,6)$. This yields a counterexample to the strongest known formulation of the Numerical Terao Conjecture.

In this paper, we investigate lower bounds on the maximum achievable service rates for data symbols in certain classes of linear codes, including cyclic codes and low-density parity-check (LDPC) codes, that are derived from combinatorial structures such as $t$-designs, difference sets, and balanced incomplete block designs (BIBDs). We first establish a lower bound on the maximum achievable service rate for each data symbol in the following two cases: (i) systematic linear codes $C$ under the assumption that the supports of codewords of a fixed weight in $C^\perp$ form a BIBD, and (ii) non-systematic binary codes under the assumption that the supports of codewords of a fixed weight in $C^\perp$ form a $t$-design. We then investigate the linear codes obtained from the incidence matrices of BIBDs, particularly certain classes of BIBD-LDPC codes, and show how the parameters of the underlying designs can be exploited to determine lower bounds on the maximum achievable service rates for the data symbols of the corresponding linear code. In addition, we analyze the maximum achievable service rates of systematic extended linear codes. We show that the existence of a symmetric BIBD (SBIBD) corresponding to a dual codeword can be used to derive a lower bound on the maximum achievable service rate of the associated systematic cyclic code. We also present some families of cyclic codes constructed from difference sets and obtain explicit lower bounds on the maximum achievable service rates for their data symbols. In particular, we determine the exact values of the maximum achievable service rates for each data symbol of cyclic codes arising from Singer difference sets.

We consider the Navier-Stokes-Korteweg equations in a bounded domain or a periodic cell. The pressure considered in this paper may not be monotone with respect to the density so that there exist non-constant equilibria allowing two-phases. Using a simple Hilbert space framework, we prove that if an isolated equilibrium is energetically stable and non- degenerate, it is exponentially stable under the isothermal Navier-Stokes-Korteweg flows when the space dimension is less than or equal to three. For non-isolated case, we prove that a global-in-time solution near an energetically stable equilibrium converges to possibly another equilibrium exponentially fast. No smallness assumptions on equilibria are imposed. For the proof we apply a (generalized) stability principle due to J. Prüss, M. Wilke and G. Simonett (2013).

In this paper, we propose a fast gradient algorithm for the problem of minimizing a differentiable (possibly nonconvex) function in Hilbert spaces. We first extend the dry friction property for convex functions to what we call the dry-like friction property in a nonconvex setting, and then employ a line search technique to adaptively update parameters at each iteration. Depending on the choice of parameters, the proposed algorithm exhibits subsequential convergence to a critical point or full sequential convergence to an “approximate” critical point of the objective function. We also establish the full sequential convergence to a critical point under the Kurdyka–Lojasiewicz (KL) property of a merit function. Thanks to the parameters' flexibility, our algorithm can reduce to a number of existing inertial gradient algorithms with Hessian damping and dry friction. By exploiting variational properties of the Moreau envelope, the proposed algorithm is adapted to address weakly convex nonsmooth optimization problems. In particular, we extend the result on KL exponent for the Moreau envelope of a convex KL function to a broad class of KL functions that are not necessarily convex nor continuous. Simulation results illustrate the efficiency of our algorithm and demonstrate the potential advantages of combining dry-like friction with extrapolation and line search techniques.