Papers

We show that a cusp surface singularity is Frobenius liftable. As a consequence, we determine all the Frobenius liftable surface singularities.

In the space $\mathbb R^3$, for Robin parameter $L>0$, it is shown that there is a family of Schrödinger operators with penetrable toroidal solenoid and variable conductivity that approximates the magnetic Aharonov-Bohm operator with a Robin boundary condition at the solenoid (border). It is also shown that approximations via smooth potentials and then a barrier in the solenoid interior give Dirichlet boundary conditions. The approximations are in the strong resolvent sense and obtained through the $\Gamma$-convergence technique, and they hold for the more general setting of smooth, closed and compact surfaces and continuous and bounded magnetic potentials.

The double lock scenario allows a secret to be transferred from one person to another without having previously exchanged secrets such as private/public keys. This can be realised by rotations of generalised calibrations in Clifford algebra. Calibrations map to Cayley-Dickson algebras and generalising this provides the mathematical structure enabling the enormous code space needed for a cryptosystem. Commuting rotations in large dimensional spaces are used but the complexity comes from the message being embedded in generalised calibrations. This system uses ideals of Clifford algebras and quasi-algebras that are sometimes embedded in Cayley-Dickson algebras making it a candidate for post-quantum cryptography. Understanding whether the new mathematics of generalised calibrations can produce a secure cryptosystem is the challenge presented in this manuscript.

We prove a Thullen-type extension theorem for hermitian bundles with Nakano semi-positive curvature. As an intermediate result we establish the $L^2$-estimates for the $\dbar$-equation in hermitian vector bundles over a non-complete Kähler manifolds. Minor corrections.

Boyd conjectured that the Mahler measure of $P_k(x,y)=x+y+\frac{1}{x}+\frac{1}{y}+k$ for $k$ an integer, is given by $r_kL'(E_k,0)$, where $E_k$ is the elliptic curve associated to the zero locus of $P_k$ and $r_k$ is a rational number. We study various arithmetic properties of $r_k$ using a dataset containing the first $250{,}000$ values of $k$, combining large-scale statistical analysis assisted by Claude with transformer-based experiments carried out using Axolver. We recover Boyd's observation that, apart from a few exceptions, $r_k$ is the reciprocal of an integer. The size of this integer is governed by the conductor of the elliptic curve. Moreover, its $p$-adic valuations display markedly different behavior according to the prime. For $p\geq 5$, the probability of $v_p(r_k)=-m$ for $m\geq 1$ appears to be $p^{-m}$. For the primes $2$ and $3$, however, we find additional arithmetic structure involving congruence conditions on $k$ and the primes of bad reduction of $E_k$. Although the neural networks do not predict $r_k$ exactly, they recover significant information about its magnitude and valuations. In particular, the experiments at the prime $2$ suggest arithmetic structure beyond the explicit predictor obtained from our statistical analysis.

Some basic properties of Ulrich modules are studied and it is proved that Ulrich modules in Cohen-Macaulay local rings are decomposable. As a result, some characterizations proved for rings having minimal multiplicity, and some properties for ideals which are Ulrich as module is discussed.

A study of the minimizers of one-dimensional Rudin-Osher-Fatemi-type functionals with linear or nonlinear total variation and fidelity term is undertaken. Conditions on the input datum and the parameters of the functional are found that force the input itself to be the minimizer or not. In the linear setting the discriminating conditions on the parameters are complementary highlighting the sharpness of the results. The nonlinear setting is substantially different: while the non-minimality of the datum is treated in analogy with the linear case, for the minimality of the input only partial answers are found. The results in the nonlinear setting hinge on auxiliary constrained or penalized minimization problems investigating the behavior of optimal transitions with prescribed height. Additionally, they are complemented by some numerical examples.

Following recent developments in numerical schemes for 3-wave kinetic equations [2, 7, 42, 44, 43], we develop spectral algorithms for multidimensional 3-wave kinetic equations and $C_{12}$ quantum Boltzmann equations with general polynomial dispersion relations. The principal numerical difficulty arises from the resonance constraint, supported on a nonlinear manifold in wave-vector space. We approximate the Dirac distribution by a truncated Fourier representation and derive two spectral discretizations of the collision operator. The first is a direct spectral method with complexity $\mathcal{O}\big(L(2N)^{3d}\big)$, while the second exploits multidimensional FFTs to reduce the complexity to $\mathcal{O}\big(L(2N)^{2d}\log(2N)\big)$. Numerical tests show excellent agreement between the two methods, with the fast algorithm providing substantial computational savings. To suppress unresolved high-frequency modes, we combine the classical $2/3$-rule with exponential spectral filtering. Simulations in two and three dimensions capture the gain–loss dynamics of the $C_{12}$ quantum Boltzmann equation for both rapidly and algebraically decaying initial data. For the 3-wave kinetic equation, the computations exhibit strong oscillations and rapid spectral broadening, providing numerical evidence of an apparent energy cascade toward high frequencies. The results also show that the dispersion relation and spatial dimension strongly influence the transient resonant dynamics.

We construct an explicit symplectic realization of the Barnes-Wall lattice that yields a family of multimode Gottesman-Kitaev-Preskill (GKP) codes with encoding rate $R = \frac{1}{2}\log_2 N$ and a deterministic $O(N\log^2 N)$ bounded-distance decoder. The recursive generator $G_{m+1} = \bigl(\begin{smallmatrix} G_m & 0 \\ G_m & R_m G_m \end{smallmatrix}\bigr)$ with $R_m = I + \Omega$ simultaneously guarantees symplectic integrality for valid quantum stabilizers and preserves the exact Barnes-Wall decoding structure through a chain of isometric isomorphisms. The code distance is constant at $\Delta^2 = 1$ (in units of $2\pi$), representing an explicit distance–rate tradeoff in which logarithmic encoding efficiency is achieved at the cost of non-scaling protection. This construction provides a deterministic, space-efficient paradigm for GKP error correction in platforms supporting non-local modular connectivity.

Mainstream plan-then-track approaches to multifingered grasp execution entail selecting a grasp, planning a collision-free trajectory, and tracking the resulting trajectory via a feedback controller. Pose-estimation error during execution or scene motion can invalidate this open-loop commitment and trigger replanning. We thus present Grasp Distance Fields (GDFs), smooth softmin distances to finite sets of arm-hand grasp configurations. Using their negative gradients as feedback, we jointly select and execute grasps without planning a trajectory. A CBF-CLF quadratic program (QP) enforces self-collision, workspace, object, and obstacle-clearance constraints, while its CLF slack quantifies obstruction of task progress. We bound the softmin approximation error by $\log N/\rho$ and prove forward invariance of the filtered safe set. To handle changes in contact topology, we combine a hysteretic contact-mode transition with a wrench-quality CBF that limits degradation of the realized force-closure margin relative to hold onset. Using our method, a fixed-base manipulator and a Unitree G1 equipped with the same underactuated hand grasp and lift 46 of 50 test objects amid clutter and moving obstacles. The realized grasps also retain a median 94% of their synthesized quality margin, and each QP solve requires 0.09 ms within a 20 ms control interval. Project page: www.clintonenwerem.com/gdf.

In the seminal paper (Yager 2015), Yager defined the negation of a probability distribution $\mathbf{p}=(p_1,\dots,p_n)$, as the distribution $\overline{\mathbf{p}} = (\overline{p}_1,\dots,\overline{p}_n)$, where $\overline{p}_i = ({1-p_i})/({n-1}),$ for $ i=1, \ldots , n.$ In this paper, we present a comprehensive information-theoretic analysis of Yager's negation and its generalizations. Using tools from information theory and majorization theory, we unify, extend, and strengthen a number of previously known properties of Yager's negation within a common framework. Overall, our results offer strong theoretical justification for Yager's negation as the most natural and principled definition of probability distribution negation under various information theoretic criteria.

In this research, a new semi-implicit two-phase double-point material point method is proposed, in which the soil and water phases are modelled using two distinct sets of material points, both being stabilised with a novel approach. The Nor-Sand constitutive model is implemented to simulate more realistic soil behaviour. Some landslide numerical examples are presented to investigate the performance of the proposed method and highlight the importance of using the double-point approach. The formulation with two sets of material points shows significantly different but more reliable results in the cases of landslides, compared with the conventional single-point approach. Furthermore, this research shows that the additional computational cost given by the additional water material points is acceptable. Therefore, it is recommended to use two sets of material points for some large deformation geotechnical problems.

In this article, we introduce the notion of connected finite graphs with disjoint cycles in normal form and show that any such graph can be transformed into a normal form graph via a finite sequence of in-splittings and out-splittings. Consequently, we provide number-theoretic criteria for meteor graphs of length three to be strongly shift equivalent, where a meteor graph of length three is a connected finite essential graph consisting of three disjoint cycles which makes a unique chain of cycles of length three. We then prove that meteor graphs of length three whose cycle lengths are pairwise coprime are shift equivalent if and only if they are strongly shift equivalent, if and only if their corresponding Leavitt path algebras are graded Morita equivalent, if and only if their graded $K$-theories, $K^{gr}_0$, are order-preserving $\mathbb{Z}[x, x^{-1}]$-module isomorphic. As a consequence, Williams' Conjecture and Hazrat's Graded Morita Equivalence Conjecture hold for graphs with disjoint cycles that contain exactly three cycles whose lengths are pairwise coprime.

We bridge two sides of singular perturbation theory: the classical theory of slow-fast systems and the semi-classical approach to quantum mechanical systems. For a specific but physically important class of dynamical systems, we show that purely classical and exotic objects, so-called canard solutions, are shadows of instantons in the corresponding quantum system. We demonstrate that canard solutions exist in a domain of parameter space whose boundaries are determined by an instanton action. We illustrate our statements analytically for the relevant example, the overdamped Josephson junction, and confirm them numerically. For the Josephson junction, the canard window is the exponentially narrow gap between consecutive Shapiro steps.

A new semi-implicit, two-phase, double-point formulation of the Material Point Method (MPM) for soil-water interaction with seepage and free-surface flows under large deformation is presented in this paper. The approach advances the water phase implicitly while keeping the soil phase explicit, enabling stable, efficient time integration in problems that involve rapid seepage and strong free-surface motion. The proposed framework models high-Reynolds-number interphase drag through a non-linear Darcy's law implemented for the first time within an incremental fractional step MPM formulation without enlarging the implicit solve. This methodology also enhances the numerical stability for fast flows and wave breaking via a hyperelastic constitutive treatment of slightly compressible viscous water, and mitigates spurious oscillations through a new stabilisation approach for the velocity. Robustness of soil-water interface is achieved by combining nodal-based, free-surface detection, suited for higher-order spline functions with smooth porosity-permeability transitions that avoid constitutive divergence at sharp material boundaries. Validation against laboratory benchmark cases reported in the literature, including pure-water dam break, dam-break seepage through a porous barrier, two granular-collapse tsunami experiments, and a dam-break wave over a movable granular bed, shows accurate and stable free-surface evolution, pressure time histories, seepage fronts, and wave-gauge records. Using an advanced critical-state soil model (NorSand) further improves the reproduction of granular flow kinematics. The results demonstrate that the proposed formulation is a reliable and computationally efficient tool for geotechnical hazards involving intense soil-water coupling, seepage, sediment transport and free water.

Subbarao and Verma introduced, in 1999, a number of open problems concerning the sequence $(f(n))_{n \geq 0}$ of complementary Bell numbers, which may be defined via Bell polynomials $B_{n}(x) = \sum_{k=0}^{n} \left\{ \begin{smallmatrix} n \\ k \end{smallmatrix} \right\} x^k$ so that $f(n) = B_{n}(-1)$. Yang [Electron. J. Combin., 2001] subsequently solved the first two problems from Subbarao and Verma, but the third such problem has remained open, to the best of our knowledge. The first part of this third problem asks whether or not $f(n)$ takes any given value only a finite number of times. We solve this problem in the affirmative, through a combined application of finite difference-based methods, partial Motzkin paths, the completeness of the Tate algebra with respect to the Gauss norm, and Strassmann's theorem.

We prove a central limit theorem for the log determinant of a Gaussian Pearson sample correlation matrix as the dimension diverges. Only two conditions are imposed: the population correlation matrix is positive definite, and the sample degrees of freedom are at least the dimension. Both are necessary for the ordinary log determinant to be finite. To the best of our knowledge, no previous central limit theorem covers this full nonsingular domain. It covers every aspect ratio from dilute growth to the square hard edge. No uniform lower or upper bound is imposed on the eigenvalues of the population correlation matrices: the smallest may approach zero and the largest may diverge. The proof develops a coordinatewise Wiener chaos reduction for the random diagonal normalization and combines it with an exact Wishart transform comparison. Geometrically, the statistic is twice the log volume of a random parallelotope spanned by standardized Gaussian coordinate vectors.

We prove the existence of infinitely many smooth nonradial stationary solutions to the surface quasi-geostrophic (SQG) equation with finite kinetic energy. Our construction is based on a family of nonradial sign-changing solutions to the two-dimensional half-Yamabe equation, obtained via a Lyapunov–Schmidt reduction and concentrated at the vertices of a regular polygon. As the number of vertices tends to infinity, the associated stationary SQG solutions converge to a radial stationary profile centered at the origin, together with a lower-order vortex sheet correction.

Height functions are fundamental and important objects and tools in mathematics, especially in geometry such as differential topology and differential geometry and some related singularity theory of differentiable maps. Our interest, especially interest of the author, lies in obtaining explicit lists of such functions. Recently, as information on so-called higher degrees, he is also interested in their naturally defined 1st derivatives. We consider natural maps on products of spheres which are variants of specific cases of so-called moment maps on toric symplectic manifolds. We also generalize cases of the canonical projections of the unit spheres. This is a further result on related previous study of the author. We use Reeb graphs, graphs being natural quotient spaces of manifolds of the domains of nice functions such as Morse-Bott functions, and consisting of connected components of level sets. They are fundamental tools and objects since the last century.

A proper conflict-free coloring is a proper vertex coloring in which every nonisolated vertex has a color occurring uniquely in its open neighborhood. We prove that every graph with neither a $K_5$-minor nor a $Q_6$-minor admits such a coloring with at most seven colors, where $Q_6=K_3\vee\overline{K_3}$. In particular, this improves the previous general upper bound of eight for planar graphs. The proof combines a previously developed iterated distance-three selector construction with a general anchor-contraction lifting principle. The first supplies independently colored witnesses in closed neighborhoods, while the second combines those witnesses with a proper coloring of a suitable minor. We also develop the parity analogue of the first mechanism and show that, whenever the $K_{k+1}$ case of Hadwiger's conjecture holds, every $K_{k+1}$-minor-free graph can be proper vertex colored with $2k-1$ colors such that every nonisolated vertex has a color occurring an odd number of times in its open neighborhood.

The divergence of Feynman graph integrals is one of the central issues in the study of perturbative quantum field theories. A rigorous formulation of these integrals usually requires renormalization. In this paper, we prove that the Feynman graph integrals arising from holomorphic field theories on closed real-analytic Kähler manifolds are convergent with respect to heat-kernel renormalization, Cauchy principal value renormalization, and zeta-function renormalization. Moreover, these three renormalization procedures produce the same value. Our proof is based on the theory of wonderful compactifications in algebraic geometry, which provides a geometric understanding of these integrals. As a consequence, we establish a gauge anomaly formula for these graph integrals.

DNA molecules are so small that it might be practical to use their frequency vectors to encode messages. More precisely, a sender can inject $M_X$ copies of the string $X =$ CATCATCAT into a pool and the receiver can recover $M_X$ by sequencing the pool. There are, however, two sources of uncertainty: (a) $M_X$ is usually too big to be counted exactly, but is estimated by sampling. (b) The DNA sequencer could be noisy; it may have difficulty distinguishing CATCATCAT from CATGATCAT. Recently, Tamir, Weinberger, and Guillén i Fàbregas clarified the amount of information the frequency vector can carry under (a). They showed that each string can carry about $\log_4 R$ bits, where $R$ is the average number of times each string is read. They also showed that $\log_4 R$ bits can be achieved by a low-complexity uncoded scheme under the condition that there are at least $\sqrt R$ distinct strings. In this paper, we show that a low-complexity coded scheme can achieve the same $\log_4 R$ bits unconditionally. We then generalize the scheme to handle sequencing noise, (b), and show that the noise penalizes the total number of bits by $\log_2 \det W$, together with a linear term due to the use of Fourier transforms in our proof. The former penalty $\log_2 \det W$ is asymptotically the same as that obtained by Gerzon, Shomorony, and Weinberger; our scheme trades a small amount of rate for practical complexity.

Bruhat decompositions give cellular models for split algebraic groups, flag varieties, and maximal compact groups, but motivic boundaries retain orientation and torus-translation data lost in the flag quotient. Over a perfect field of characteristic zero, let the group be connected, split, semisimple, and simply connected. Fixing a Borel subgroup with split maximal torus and unipotent radical, we construct a torus-enriched motivic cellular complex for the basic affine space and compute its boundary in every degree. Each cover in Bruhat order contributes a two-face operator determined by a transported coroot, a tail determinant weight, and an explicit Milnor–Witt frame degree. Bott–Samelson purity proves the formula, while the unipotent torsor identifies the complex with that of the group. Over the real numbers, realization identifies it at chain level with the extended-Weyl complex of a maximal compact subgroup, while torus augmentation gives the flag complex. A single motivic complex therefore interpolates between the two incidence theories. A finite torus-support filtration makes this explicit; after inversion of two it splits by the characters of the component group of the real split torus, and the support spectral sequence degenerates. Calculations in the rank-three special linear and exceptional rank-two cases exhibit the first higher differentials beyond the previously known range.

A directed triple system can be defined as a decomposition of a complete digraph to directed triples $\langle x,y,z\rangle$. By setting $xy =z$, $yz =x$, $xz =y$ and $uu =u$ we get a binary operation that can be a quasigroup. We give an algebraic description of such quasigroups, explain how they can be associated with triangulated pseudosurfaces and report enumeration results.

It has remained open to identify the necessary and sufficient conditions for the $L^2$ convergence of hyperinterpolation since it was introduced by Sloan in 1995. We show that the $L^1$-$L^2$ Marcinkiewicz-Zygmund (MZ) condition, together with the asymptotic functional approximation property for polynomials, is the answer. We further prove that the optimal $L^1$-$L^2$ MZ constant coincides with the operator norm of the hyperinterpolation operator, and it admits a natural Banach space duality interpretation. With an explicit construction, we also show that Pólya's classical conditions for quadrature convergence are not sufficient for the $L^2$ convergence of hyperinterpolation. This reveals a fundamental distinction between the convergence of linear functionals (quadrature formulas) and that of linear operators (hyperinterpolation operators). We establish a strict logical hierarchy for the stability and accuracy conditions governing the convergence of quadrature and hyperinterpolation.

In this paper, by using $q$-identities and differential operator techniques, we establish two $q$-supercongruences modulo the square of a cyclotomic polynomial, which are associated with truncated double basic hypergeometric $q$-series.

Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F}_q$, and ${V\brack k}$ denote the family of all $k$-dimensional subspaces of $V$. The families $\mathcal{F}\subseteq {V\brack k}$ and $\mathcal{G}\subseteq {V\brack \ell}$ are said to be cross $t$-intersecting if $\dim(F\cap G)\geq t$ for all $F\in\mathcal{F}$ and $G\in \mathcal{G}$. In this paper, we determine the extremal structures when $|\mathcal{F}||\mathcal{G}|$ attains the maximum value under the conditions $\dim\left(\cap_{F\in \mathcal{F}}F\right)<t$ and $\dim\left(\cap_{G\in \mathcal{G}}G\right)<t$.

We study the separability of subsets of an infinite group. Given an infinite group $G$ and subsets $A,B\subset G$, we say that $A$ and $B$ are separated in $G$ if there exists an infinite symmetric subset $X\subset G$ with $e\in X$ and $XAX\cap B=\varnothing$. Question 17.102 of the Kourovka Notebook raises a natural cardinal question: if $A$ and $B$ are disjoint and $|A|,|B|<|G|$, must they be separated? We first give a negative answer, constructing two essentially different families of counterexamples, based respectively on rigid binary relations and on torsion-free groups whose square sets have cardinality smaller than that of the group. On this basis we introduce the $W$-witness set $W_G(A,B)=\{g\in G:\{g,g^{-1}\}A\{g,g^{-1}\}\cap B=\varnothing\}$, obtain a necessary and a sufficient condition for separability given by its size, and study the critical range. The central result of this paper is a necessary and sufficient characterization of separability: taking the inverse pairs $\{g,g^{-1}\}$ as vertices, we construct the conflict graph $\Gamma_{A,B}$, and $A$ and $B$ are separated if and only if the associated graph contains an infinite independent set; separability is thereby converted into an ordinary graph-theoretic problem. From this we further derive the case of finite subsets, the case of Abelian groups, and several cardinal criteria, and we obtain, via Ramsey's theorem, a structural dichotomy for the non-separable case. In addition, we study the range of cardinalities of counterexample groups and the possible sizes of $W$-witness sets, and prove that no uniform necessary and sufficient criterion can depend only on $|A|$, $|B|$ and structural invariants of the group, independently of the specific position of $A$ and $B$.

We prove sequential convergence results for the AdaGrad algorithm family optimizing convex differentiable objectives. Specifically, we provide necessary and sufficient conditions for the convergence of iterates for the three main AdaGrad variants (AdaNorm, AdaDiag, AdaFull) when the objective is convex and locally Lipschitz-smooth, closing the question left open from the literature. We harness this general result to study the three variants under the generalized $(L_0,L_1)$-smoothness condition and show sequential convergence for sufficiently small constant step size. Moreover, under the so-called $(L_0,L_1)$-polynomially modifiable smoothness assumption, which is a relaxation of the $(L_0,L_1)$ generalized smoothness property and is satisfied by many function classes such as $L$-smooth functions or univariate polynomials, sequential convergence for these AdaGrad variants is proved for arbitrary learning rates. This result provides conditions under which AdaGrad presents adaptivity, i.e., does not require tuning the parameters based on the instance. Finally, we provide numerical illustrations of the behavior of AdaGrad on convex and nonconvex functions. In particular, we construct a counterexample empirically showing that smoothness alone is not sufficient for the sequential convergence of AdaGrad-type algorithms, and suggesting that additional geometric hypotheses (e.g., convexity as in this paper, or the Kurdyka-Łojasiewicz inequality) are indispensable for sequential convergence results.

It is well known that the number of partitions of $[n]$ without singletons equals the number of partitions of $[n]$ in which no block contains two cyclically adjacent elements $i,i+1\pmod{n}$. Bernhart remarked that there might be no simple bijection between these two classes. Although Callan later constructed an algorithmic bijection proving the stronger equidistribution of singletons and adjacencies, his construction proceeds through multiple rounds of exchanges. Therefore, Bernhart's remark may still retain some validity, as suggested by Chen and Wang. In this article, we address this remark by giving a direct “one-round” bijection between the two classes. Unlike Callan's bijection, our map is closely compatible with the refinement order on partitions: in one direction it only decomposes blocks, while its inverse only merges blocks, with a single exceptional pair when $n>2$ is even. We further observe that this exception is unavoidable, establishing the optimality of the bijection with respect to the refinement order. The specific local form of these operations — splitting off only singleton blocks and merging a singleton only with the block containing its cyclic neighbor — also ensures that the construction restricts, without modification, to a bijection between the corresponding classes of noncrossing partitions.