Papers

For an integer-valued sequence $\{a(n)\}_{n\geq 0}$ and a prime $\ell$, a Ramanujan-type congruence is a relation of the form $a(\ell n-\delta_{\ell})\equiv 0\pmod\ell$, where $\delta_{\ell}$ is a specific shift. In this paper, we present explicit constructions of modular forms to establish Ramanujan-type congruences for a broad class of generating functions, including eta-quotients, weakly holomorphic modular forms, and mock modular forms. As applications, our explicit approach provides a unified framework to not only recover known congruences but also establish new non-congruence results and explicit congruences for various combinatorial and arithmetic functions.

This note addresses the resonance of monomial subspaces. In the first part, we completely characterize the cases where the Fitting scheme structure of the resonance associated with a monomial subspace is reduced, and we further investigate the primary decomposition of the corresponding Fitting ideal. The second part focuses on non-monomial subspaces having the resonance of a monomial subspace.

Using the energy function on the extended Siegel-Jacobi upper half space of order $n$, $\tilde{\mathcal{X}}^J_n$, with $n\in \mathbb{N}$, the equations of motion in the variables $(x,y,q,p,\kappa)$ attached to linear Hamiltonians in the generators of the real Jacobi group $G^J_n(\mathbb{R})$ are presented, where $x,y$ are symmetric matrices in $\mathcal{M}(n,\mathbb{R})$ and $p,q$ are real $n$-vectors. The case $n=1$ is presented separately.

We formulate Schneider–Teitelbaum duality between wide classes of Banach $k$-linear representations of $G$ and left $O_k[[G]]$-modules for a non-spherically complete field $k$, e.g.\ $\mathbb{C}_p$, and a profinite group $G$. We interpret a topological notion of a weak variant of irreducibility of a Banach $k$-linear representation of $G$ into a purely algebraic notion of a certain simplicity of the dual left $O_k[[G]]$-module. As applications, we give two $p$-adic families of infinite dimensional Banach $\mathbb{C}_p$-linear representations of a $p$-adic Lie group satisfying the weak irreducibility.

We present a novel multi-modal optical coherence photoacoustic elastography framework, which combines two imaging modalities, optical coherence tomography and photoacoustic tomography, to enable complementary absorption-scattering measurements for the extraction of quantitative tissue features via quasi-static elastography. For this, we develop a sophisticated hybrid inversion algorithm for merging the complementary information layers contained in both optical coherence and photoacoustic-based elastography measurements, and perform systematic evaluations to assess the impact of hybrid elastography data on strain and stiffness reconstructions. An extension to a photoacoustic tomography and optical coherence tomography imaging system allows precise elastographic experiments. Studies on a silicone elastomer phantom demonstrate that the combined optical coherence - photoacoustic approach outperforms single-modality optical coherence elastography and photoacoustic elastography, yielding higher strain signal-to-noise ratios and improved stiffness estimates. These results establish the advantage of multi-modal complementary imaging and data merging for accurate, high-resolution elastographic strain and stiffness mapping in both scattering and absorbing materials.

Most currently known kissing arrangements of size $840$ in $\mathbb R^{12}$ share a common structure. They consist of $60$ vectors supported on $\mathbb R^6\times\{\mathbf 0\}$, another $60$ vectors supported on $\{\mathbf 0\}\times\mathbb R^6$, and $720$ additional bridge vectors. The bridge vectors encode the interaction between the two six-dimensional factors and are constructed from the unique $1$-factorization of the complete graph $K_6$. In this paper we investigate kissing arrangements of this type while keeping the bridge vectors fixed. We show that each $60$-point block admits substantial flexibility: $12$ of its vectors may be chosen as the signed coordinate vectors $\pm e_i$, while the remaining $48$ vectors may vary within a positive-dimensional family of configurations, which we call $48$-systems. As a consequence, we obtain infinitely many pairwise non-isometric kissing arrangements of size $840$ in $\mathbb R^{12}$. The geometric freedom revealed by these constructions provides new insight into the local structure of extremal configurations. Exploiting this structure, we develop a specialized initialization scheme for logarithmic Riesz energy optimization. Starting from such structurally informed initial configurations, we numerically construct a kissing arrangement of size $841$ in $\mathbb R^{12}$.

Thermal runaway remains a critical safety concern in battery systems and other thermally active devices, necessitating reliable methods for predicting the conditions under which thermal runaway may occur. In this work, we develop a finite element framework for assessing thermal stability in systems undergoing transient heat conduction with temperature-dependent internal heat generation. The formulation leads naturally to a generalized eigenvalue problem, wherein the sign of the smallest eigenvalue provides a direct criterion for determining the onset of thermal runaway. This approach enables direct prediction of thermal runaway thresholds in geometrically complicated problems without requiring computationally expensive transient analysis. The proposed methodology is validated through comparison with analytical solutions for a cylindrical Li-ion cell, as well as with a pack of cylindrical cells, demonstrating excellent agreement in both cases. Based on this model, the influence of material properties, boundary conditions, geometric parameters, and spatially varying heat generation on stability limits is examined. A key advantage of this formulation is its effectiveness even for complex geometries where analytical methods become impractical. By providing a systematic and computationally efficient means to identify stability thresholds, the present work offers a practical tool for the thermal design and safety assessment of battery systems.

We prove a local exact controllability result for controlled Landau–Lifshitz–Gilbert equations on $\mathbb T^2$: if the initial energy is sufficiently small, then for any terminal time $T>0$, there is a localised external magnetic field such that the system can be steered exactly to the terminal value of any nearby uncontrolled trajectory. We first transform the equation to a quasilinear parabolic system on $\mathbb R^2$ by a suitable stereographic chart. Then the Carleman estimate is established for the linearised system through a decomposition adapted to the self-adjoint and skew-adjoint structure of the conjugated adjoint operator. This yields observability and $L^\infty$-null controllability for the linearised system. The nonlinear projected equation is then recovered by a Kakutani fixed-point argument. We also obtain a semi-global controllability result under a hemisphere condition.

In a previous note, a two-player game between a random player and a deterministic player was introduced, and it was shown analytically that the winning probability of the deterministic player is the derangement probability dn/n!. The natural question left open was to explain this coincidence combinatorially. This paper gives such an explanation and extends it to a larger family of games. In the generalized game, the deterministic player removes q elements per turn, or all remaining elements if fewer than q remain. We couple the game exactly to the cycle decomposition of a uniformly random permutation. Under this coupling, the random player wins precisely when the first cycle of length at most q, read in canonical cycle order, is a fixed point. The case q = 1 recovers derangements, while the general case is governed by the first short cycle of the permutation.

Response-adaptive randomization designs have attracted much attention in clinical trials. This paper proposes a new class of response-adaptive design which built on the efficient randomized-adaptive design (ERADE) proposed by Hu, Zhang, and He. The original ERADE uses a discrete allocation probability function and leverages the stopping time theory of stochastic processes to establish asymptotic results. It has been proven that the design can reach the Cramér-Rao lower bound for any target allocation proportion. This study further expands the original framework by replacing the traditional case-by-case sequential enrollment with group recruitment over fixed time intervals(weekly, biweekly, or monthly), and dynamically updates the allocation probabilities based on the cumulative response information within each group. Meanwhile, to better fit practical application scenarios, we further explicitly consider the situations of randomly missing data and response delay. Theoretical analysis shows that the new design retains all the main asymptotic properties of the original ERADE, and still performs well under the conditions of response delay and missing data. Finally, through simulation studies and the redesign of a real-world clinical trial, the effectiveness and practicality of the proposed method are verified.

Since every modeling process in real-world situations is subject to errors, the study of the so-called shadowing property becomes highly relevant. This property allows for the identification of true orbits that closely follow a chain of trajectories in which some degree of approximation is introduced at each step. The objective of this paper is to establish a version of the Shadowing Theorem for vector fields governed by ordinary differential equations in the piecewise-smooth setting, where the dynamics alternate between distinct regimes, or on-off stages. First, we present an example showing that such a result can not be obtained under the same hypotheses commonly assumed in the C2 scenario. Consequently, under appropriate assumptions, we prove a Shadowing-like Theorem specifically tailored to this framework. Furthermore, we propose several extensions of the main result aimed at capturing characteristic phenomena intrinsic to piecewise-smooth vector fields - features that do not arise in the C2 setting.

We extend the kernel-based symplectic predictor of [1] to a parameter-augmented setting in which the learned flow-map surrogate depends not only on the state, but also on additional variables such as physical parameters and macro time-step sizes. The method uses a product kernel ansatz on a parameter and macro step augmented domain and constructs the prediction through an implicit symplectic-Euler-type update. Hence, for every fixed admissible parameter and time-step instance, the resulting large-step predictor is symplectic by construction. The training problem is formulated as gradient Hermite–Birkhoff interpolation in a reproducing kernel Hilbert space. Efficient surrogates are obtained by greedy center selection. We show that the convergence analysis from the non-augmented setting carries over to the product-kernel framework and derive corresponding prediction error bounds. Numerical experiments for a pendulum with varying length and time-step size and for a parameter-dependent discretized wave equation illustrate the accuracy and structure-preserving behavior of the proposed approach.

This paper studies the optimal portfolio, consumption, and endogenous early retirement problem within a benchmark tracking framework by incorporating a new relative performance evaluation. In this framework, the investor maximizes expected lifetime consumption utility while managing the maximum wealth shortfall relative to a benchmark, with shortfall-management costs that may differ before and after retirement. Mathematically, the problem is a hybrid stochastic control problem involving both regular controls and an optimal stopping time, in which the running maximum process records the investor's largest benchmark shortfall. We introduce an auxiliary reflected state process and establish an equivalent hybrid stochastic control problem. By proving the convex duality theorem, we technically transform the original problem into a two-dimensional pure optimal stopping problem with state reflection. This enables us to characterize the geometric structure of the stopping set and derive the feedback-form optimal retirement boundary, as well as optimal portfolio and consumption policies. Analytical examples and numerical simulations reveal a two-stage structure with more conservative investment and more aggressive consumption after retirement. Driven by the retirement option, the expected largest shortfall risk follows a pronounced U-shaped pattern with respect to wealth. Shortfall management costs, labor income, and leisure preference significantly influence retirement timing, investment, and consumption.

We study monadic dependence of binary relational structures including at least one antisymmetric relation. Our cornerstone result gives sufficient conditions for proving that a structure is monadically dependent by only considering some of its reducts, assuming they are structurally well-behaved and compatible enough. As an application, we consider some reorientation rules preserving monadic dependence of binary structures, as well as replacement of one antisymmetric relation with bounded independence number by another. Then, we apply our main (technical) result to the study of twin-width in two ways. First, generalizing the fact that twin-width boundedness is equivalent to being expandable by a linear order into a monadically dependent class, we prove that it is also equivalent to being expandable by an oriented graph with bounded independence number (for instance by a poset with bounded width or by a tournament), and that FO-model checking is fixed-parameter tractable on such an expansion. Second, we show delineation by twin-width for some new classes of oriented graphs, including oriented split graphs and local tournaments. In all these cases, we also obtain fixed-parameter tractability of FO-model checking.

We study the angle between the null space and the range of elementary operators of length one or two acting on $\mathcal{B}(\mathscr{X})$, the Banach algebra of all bounded linear operators on a complex Banach space $\mathscr{X}$. For the multiplication operator $\mu_{A,B}(X) = AXB$, we characterize positivity of this angle in terms of the corresponding angles for $A$ and $B^*$. For elementary operators of length two $\Delta_{\boldsymbol{A},\boldsymbol{B}} = \mu_{A_1,B_1} - \mu_{A_2,B_2}$, we establish conditions under which the angle is positive, and the ascent of $\Delta_{\boldsymbol{A},\boldsymbol{B}}$ equals one. Finally, for a generalized derivation $\delta_{A,B}$ and an injective holomorphic function $f$ on a neighborhood of $\sigma(A)\cup\sigma(B)$, we show that the angle between the null space and the range of $\delta_{f(A),f(B)}$ is positive whenever the angle between the null space and the range of $\delta_{A,B}$ is positive.

We introduce dynamical blanket times, which quantify how quickly the empirical distribution along a typical orbit approximates the invariant measure. These can be viewed as a measure-theoretic analogue of the previously introduced dynamical cover time, which measures how quickly an orbit becomes dense in the space. Motivated by analogous comparability questions for random walks on graphs, we investigate how much longer it takes for a dynamical system to “blanket” than to “cover”. For finite-branch, uniformly expanding interval maps, we obtain upper bounds on the expected blanket time in terms of the spatial scale and the precision of approximation. In the special case where the invariant measure is absolutely continuous with respect to Lebesgue, this yields comparability between the expected blanket and cover times, uniformly across all sufficiently small scales. Our approach combines two main ingredients. First, we establish large deviation estimates for hitting times which are uniform over both the target location and the spatial scale. Second, using methods from multifractal analysis, we construct a finite discretisation of the invariant measure which reduces the problem to a suitably controlled discrete model.

Euclidean lattices are an interesting object of study in many regards and can have a rich structure arising from various constructions, e.g., from number field extensions. A particularly interesting class is the one of well-rounded lattices, as they relate to the well-known densest sphere packing problem in geometry, theta function minimization, and the famous Minkowski and Woods conjectures. In addition to being an important mathematical object in their own right, lattices also play a central role in many applications. This paper offers a survey of structured lattices and discusses their recent applications in lattice-based cryptography and secure wireless communications. Our goal is to spark the interest of mathematicians and adjacent communities in these fascinating topics in the intersection of lattices, number theory, cryptography, and wireless communications.

We review the probabilistic representation of solutions of wave equations with polynomial nonlinearities in spatial dimensions d=1,2,3 using stochastic branching processes. Under regularity assumptions on the initial data, we derive conditions ensuring the integrability of the corresponding Monte Carlo estimator, and the existence and smoothness of mild and classical solutions. We also present numerical results and comparisons with grid-based algorithms for the solution of nonlinear wave equations.

We introduce a rank-one Riemannian cometric update inducing a modification of the Riemannian metric that makes specific directions of motion cheaper to travel along. We establish basic completeness properties of this reward metric, and give an explicit characterization of its Levi–Civita connection. We propose a preconditioned trajectory-tracking strategy by adding the connection-difference term to a standard intrinsic PD control, and illustrate the construction on a connection control-affine system on the Special Euclidean group with a maze navigation experiment. When the nominal trajectory is an integral curve of the vector field used to define the reward metric, our methodology improves the overall tracking, which is demonstrated through simulation results.

This paper constructs a connection map on the second-order tangent bundle induced by a linear connection on the base manifold and uses it to define a generalized Sasaki metric. The associated geodesic equations are derived, and jet-constrained variational problems are shown to yield Riemannian quintics in tension. The construction is then specialized to rigid body attitude dynamics with first-order actuator dynamics, producing an intrinsic higher-order trajectory model on the rotation group. Numerical simulations compare quintics in tension with Riemannian cubics as nominal trajectories and show modest reductions in actuator-relevant cost with comparable tracking performance.

Duality relations for hypergeometric functions have reappeared as an active research topic several times, with the first instances tracing back to Euler and Gauss and the latest burst of activity occurring between 2015 and 2023. In this paper we present a common generalization of all relations of this type found in the existing literature both for hypergeometric and for $q$-hypergeometric functions. We cover both Gauss type and confluent generalized hypergeometric functions and their $q$-analogues. Our results entail a number of corollaries including multi-term relations for hypergeometric and $q$-hypergeometric series at a fixed argument.

Environment division multiple access (EDMA) exploits line-of-sight (LoS) availability, blockage diversity, and spatial isolation in the propagation environment to regulate inter-region interference, while non-orthogonal multiple access (NOMA) improves intra-region access efficiency through power-domain multiplexing and successive interference cancellation (SIC). This paper investigates heterogeneous environment-map-aware user matching and power allocation for pinching-antenna-assisted EDMA-NOMA systems. Based on large-scale path loss and an exponential LoS blockage model, an average LoS/NLoS effective large-scale channel gain is constructed, allowing the environment map to affect both desired service links and inter-region interference links. This avoids the overly optimistic assumption that environmental blockage only suppresses interference. A utility-based joint user-pairing and power-allocation algorithm (UBA-JPPA) is then proposed to jointly account for system throughput and the scheduled-user Jain fairness index under NOMA power constraints and an SIC-consistent rate definition. Simulation results show that, compared with the ablation baseline without a heterogeneous environment map, the proposed scheme provides consistent gains in sum rate, fairness, and the throughput–fairness operating region. Meanwhile, a strong EDMA-OMA benchmark remains competitive in some high-SNR regimes, indicating that the focus of this work is to verify the benefit of heterogeneous environment information for EDMA-NOMA resource allocation rather than to claim that NOMA universally outperforms OMA.

An ideal in a polynomial ring is symmetric if it is invariant under any permutation of variables. In this paper, we define and study the symmetric and isotypic Hilbert series for symmetric ideals in a polynomial ring with countably many variables. The symmetric Hilbert series is the limit of the Hilbert series of the invariant parts of the finite truncated quotients, while the isotypic Hilbert series records stable multiplicities of irreducible symmetric-group representations for each degree. Our main result proves that, under a mild support condition on the ideal, the symmetric Hilbert series is a rational function. We further show that this rationality extends to the isotypic Hilbert series for every irreducible representation. The proofs of these results rely on the monomial structure of the polynomials within the symmetric ideal, combined with Kostka inversion for the isotypic case.

The study of word maps and Waring-like problems has been widely pursued for finite simple groups, algebraic groups, and Lie groups. In this article, we study Engel word maps $e_{m}(x, y) = \left[\cdots\left[[x, y], y \right], \cdots, y \right]$ on certain linear groups over local rings, namely, $\mathrm{SL}_2(\mathcal R)$ and $\mathrm{PSL}_2(\mathcal R)$. We consider the commutative ring $\mathcal {R} $ to be either a complete, local principal ideal ring $\mathcal O$, or a local principal ideal ring of finite length $\mathcal O_\ell$. Suppose the characteristic of the residue field $k\cong \mathbb F_q$ is $\neq 2$. Under some mild conditions on $q$, we show that there exists a constant $q_0(m)$, such that for all $q \geq q_0(m)$, all lifts in $\mathrm{SL}_2(\mathcal{O})$ of non-scalar elements of $\mathrm{SL}_2(k)$, are in the image of the $m$-th Engel word over $\mathrm{SL}_2(\mathcal{O})$. We further show that all Engel word maps are surjective on $\mathrm{PSL}_2(\mathcal{O}_2)$ where $\mathcal{O}_2$ is a local principal ideal ring of length two. This work generalizes similar results about the Engel word map over fields.

For compressible nonlinear isotropic elasticity it is well known that rank-one convexity, polyconvexity and the monotonicity of the Cauchy stress tensor with respect to the logarithmic stretch tensor (the true-stress-true-strain monotonicity, TSTS-M$^+\!$) are independent constitutive conditions which should, however, all together be satisfied for a physically meaningful description of idealized elastic materials. In the incompressible case, TSTS-M$^+\!$ turns into Hill's inequality since the Cauchy stress $\boldsymbol{\sigma}$ reduces to the Kirchhoff stress $\boldsymbol{\tau}$. Hill's inequality requires then monotonicity of the Kirchhoff stress in terms of the logarithmic stretch tensor evaluated for incompressible response. In this paper we clarify how the a priori independent notions of Legendre-Hadamard ellipticity (LH), polyconvexity and Hill's inequality are nevertheless intimately connected. More precisely, by providing several alternative proofs, we show that both LH-ellipticity (rank-one convexity) and polyconvexity imply the weak Hill inequality in the incompressible two-dimensional case.

The present paper considers McKean-Vlasov SDEs with density-dependent spatially unbounded drift, which may be viewed as a non-linear density-dependent perturbation of the Ornstein-Uhlenbeck process. We develop a comprehensive theoretical framework for this class of equations. First, we establish strong well-posedness and derive optimal Gaussian pointwise bounds for both the solution density and its gradient. Then we derive an explicit expression for the stationary density and show that it satisfies logarithmic Sobolev and Poincaré inequalities. Finally, we prove exponential convergence to equilibrium in the \(\chi^2\)-metric.

Let $x$ be uniformly distributed on $(0,1)$, and let $(q_n)_{n\geq1}$ be the digits of its Engel series expansion. We establish a Cramér-type moderate deviation expansion for $(\log q_n-n)/\sqrt n$. The proof is based on a martingale decomposition and asymptotic results for martingales. As consequences, we obtain a moderate deviation principle over the full range of scales between the central limit theorem and the law of large numbers, without the additional lower rate restriction required in several earlier works. We also derive a uniform Berry–Esseen bound of order $(\log n)/\sqrt n$.

This paper proposes a novel approach to design of Nonlinear Model Predictive Control (NMPC) schemes based on Finite-Gain Stability (FGS) concepts. The proposed formulation considers the case where the plant is affected by unknown but bounded disturbances, which renders difficult the classical Lyapunov-based analysis/design. Based on FGS conditions for a closed-loop system, we develop a systematic NMPC design methodology, allowing us to choose the relevant NMPC parameters that lead to closed-loop FGS and provide a satisfactory tracking performance, also for the case of time-varying reference signals. A simulated example is presented to demonstrate the effectiveness of our framework, concerned with lateral/longitudinal control of an automated vehicle.

We study the parametric subfamily $p = 3m(m+1) + 1$ with $m = 2^a 3^b - 1$, $a,b \in \mathbb{N}^*$, a 3-smooth slice of the centred hexagonal numbers $3m^2 + 3m + 1 = (m+1)^3 - m^3,$ from the point of view of unconditional primality certification via the Pocklington-Lehmer criterion. The 3-smoothness of $m+1 = 2^a 3^b$ yields, for every $(a,b)$, a fully factored divisor $F = 2^a 3^(b+1)$ of $p-1$ satisfying $F > \sqrt(p)$ unconditionally, reducing the certificate to two witnesses, for $q = 2$ and $q = 3$. Our main new contribution is a complete, deterministic characterisation of the two canonical witnesses. We prove that $w_2 = 5$ is a valid witness if and only if $a - b$ = 1, 2 (mod 4), by quadratic reciprocity; and that $w_3 = 7$ is a valid witness if and only if $m$ is not congruent to 2 (mod 7), by cubic reciprocity in $\mathbb{Z}[omega]$ using the explicit Eisenstein factorisation $p = ((1+m) - m \omega)((1+m) - m \omega^2)$. These two results turn the heuristic "5 and 7 always work" (which is in fact false) into exact congruence conditions, and yield a deterministic witness-selection rule. Alongside, three elementary arithmetic filters (mod 6, a (-3) quadratic-residue sieve, and a mod-7 forbidden-class test) remove about 87% of candidates at negligible cost. As a demonstration, a multi-core implementation produced four unconditional certificates on consumer hardware, the largest a prime of 29998 decimal digits.

Linear problems often include many parameters that may be uncertain. Sensitivity analysis studies how these parameters impact optimal values. Instead of analyzing the objective function, we shift the focus to the optimal values of the variables. Three types of linear modifications are considered: on the cost vector, the right-hand side, and on the constraint matrix. Several theorems establish properties of these modifications, including conditions for continuity of optimal variable values, as well as local convexity and concavity properties.