Papers

Quantum Markov models have been successfully used to accurately model various physical quantum systems in fields such as quantum optics, optomechanics and superconducting circuits and they provide the basis for (measurement-based) quantum feedback control. However, the quantum Markov assumption is a strong one and it is not expected to hold for general quantum systems of interest. The projection operator approach is one approach that has been developed to model non-Markovian quantum systems by considering its embedding in a larger Markovian quantum system, but mainly in the context of quantum master equations for the dynamics of the unmonitored reduced quantum state of a quantum system. This approach was recently adapted for continuously measured non-Markovian quantum systems, which enables open-loop control but did not yet consider the presence of feedback of the stochastic measurement record, deriving non-Markovian SDEs for the evolution of the projected state of the Markovian embedding. This paper generalizes these stochastic equations to the setting of stochastic feedback based on the continuous-measurement record and shows that the equations take the same form but that previously deterministic terms become stochastic ones which depend on the measurement record, as would be intuitively expected. The stochastic equations are obtained for a generalized class of measurements that includes continuous (possibly adaptive) homodyne and photon counting measurements.

Let $X$ and $Y$ be locally compact Hausdorff spaces. We study order isomorphisms \[ T:C_0^+(X)\to C_0^+(Y), \] where $C_0(X)$ denotes the Banach space of all real-valued continuous functions on $X$ vanishing at infinity, and \[ C_0^+(X)=\{f\in C_0(X):f\ge0\} \] is its positive cone. We assume that $T$ is positive homogeneous. That is, \[ T(rf)=rT(f) \qquad (r>0,\,f\in C_0^+(X)). \] Under this assumption, we prove that $T$ is represented as a weighted composition operator induced by a homeomorphism from $Y$ onto $X$ and a bounded continuous weight function. Moreover, we show that $T$ extends uniquely to a linear order isomorphism between $C_0(X)$ and $C_0(Y)$.

This paper is dedicated to Paul Butzer on the occasion of his 85th birthday. His work and example have strongly influenced not only the first author, but also generations of mathematicians working in approximation theory and Fourier analysis. He has shown younger colleagues the importance of remaining open to applied areas, avoiding an overly narrow scope, and exploring different ways of understanding mathematical facts. A recurring theme in his work is the logical equivalence of fundamental statements in analysis. It may be less widely known that, besides his central role in approximation theory, Paul Butzer has also made significant contributions to probability theory. We hope that he will enjoy this note, which shows that a purely functional-analytic treatment of generalized stochastic processes is possible. The approach is based on the Segal algebra S0(G) and avoids several technical difficulties associated with the customary framework of vector-valued integration and topological vector spaces.

We consider classical 2d lattice models with face interactions defined in terms of a fusion category. The symmetries of such models typically include an algebra of topological operators sitting on a closed path in the lattice. In the case when the face interactions obey the Temperley-Lieb (TL) relations, we present a generic algorithm to determine the decomposition of the transfer-matrix space of states as a direct sum of simple TL modules. We apply this approach to several examples, and analyse the action of topological operators. As an application, we compute the modular transformation of the irreducible TL characters at primitive roots of unity.

We provide an upper estimate à la Pleijel on the asymptotic number of nodal domains for eigenfunctions corresponding to the cogenus eigenvalues $\{\lambda_k(p;\Omega)\}$ of the $p$-Laplacian in a bounded domain $\Omega$, and identify regimes when the number of nodal domains of the $k$-th eigenfunction is less than $k$ as $k \to +\infty$. As auxiliary results, which also have independent interest, we provide a useful characterization of the cogenus eigenvalues implying their continuity with respect to $p$, justify the Weyl law, and prove the inequality $\lambda_2(p;B) \leq \dots \leq \lambda_{N+1}(p;B) \leq \lambda_\ominus(p)$ in an $N$-dimensional ball $B$, where $\lambda_\ominus(p)$ is an eigenvalue whose eigenfunction has a central section of $B$ as its nodal set.

We study a quantized multiple-input multiple-output (MIMO) integrated sensing and communication (ISAC) system in which the communication and sensing receivers each apply analog spatial combining followed by scalar subtractive dithered quantization. This quantization model leads to an additive effective-noise representation with non-Gaussian noise. We derive upper and lower bounds on the capacity of this channel. Numerical results show that these bounds are tight at low signal-to-noise ratios (SNR) and saturate at high SNR due to finite-resolution quantization. They also show that, despite the effective noise being non-Gaussian, independent and identically distributed (i.i.d.) isotropic Gaussian signaling achieves rates close to capacity. Focusing on i.i.d. Gaussian signaling, this paper also presents a closed-form expression for the linear minimum mean-squared error (LMMSE) achieved under a Kronecker sensing-channel model. Numerical results show that the LMMSE also saturates at high SNR, where the saturation level increases as the spatial combining ratio decreases, and for combining ratios below one, saturation occurs even without quantization.

We present a comparison principle for unbounded viscosity solutions to Hamilton-Jacobi equations on Wasserstein spaces of probability measures over $R^d$ . In addition to the use of standard techniques of viscosity solutions, our approach requires a key extension on Wasserstein spaces of a result of perturbed optimization on Banach spaces due to Stegall.

We derive sufficient and almost necessary conditions for large sample and catalytic majorization between finite statistical experiments over standard Borel sample spaces. This work generalizes previous results, on one hand, in the bivariate case and, on the other hand, in the multivariate discrete (or, rather, finite) case, i.e., matrix majorization. We derive multivariate generalizations of the bivariate Renyi relative entropies and show that inequalities involving these multivariate Renyi divergences characterize large-sample and catalytic majorization of finite statistical experiments. As our methods are real-algebraic in nature, this work demonstrates that large deviation techniques are not the only option available to derive conditions for large sample majorization even in the case of more general sample spaces of the experiments. We also show that all general multivariate divergences, i.e., multivariate extensive and monotone maps of finite statistical experiments, can be expressed through barycentres over the set of multivariate Renyi divergences. We also show that we may characterize the optimal conversion rate of a statistical experiment into another using the multivariate Renyi divergences.

A diffeomorphism of a $4$-manifold is said to be exotic if it is continuously isotopic to the identity but not smoothly isotopic to the identity. Ruberman constructed the first examples of exotic diffeomorphisms on simply-connected closed $4$-manifolds. His examples were reducible $4$-manifolds that necessarily have even $b_+$ in order that they can be detected by the families Seiberg–Witten or Donaldson invariants. Later Konno and Baraglia produced exotic diffeomorphisms on irreducible $4$-manifolds with odd $b_+$. In this paper, we will construct exotic diffeomorphisms on reducible $4$-manifolds with odd $b_+$. Exoticness is detected using a families Bauer–Furuta invariant. In proving our results we need to work with families moduli spaces which are not framed and so do not give rise to framed cobordism invariants. We overcome this difficulty by considering a Bauer–Furuta type invariant valued in pin-cobordism. In addition to constructing exotic diffeomorphisms, we also find new examples of simply-connected $4$-manifolds whose mapping class groups are not finitely generated.

Self-dual cyclic codes have garnered significant interest owing to their rich algebraic structures and wide-ranging applicability. Their construction and the establishment of lower bounds on their minimum distances are fundamental problems in coding theory. Chen and Ding laid an important foundation for the construction of self-dual cyclic codes in the case where the multiplicative order of $q$ module $n$, denoted by $\operatorname{ord}_n(q)$, is odd. Building on their work, we extend the investigation to the case of even order $\operatorname{ord}_n(q)$ and demonstrate that the minimum distances of the resulting self-dual cyclic codes satisfy square-root lower bounds. By examining the consecutive zero segments in the defining set of the dual code, we determine the exact parameters of Euclidean self-dual cyclic codes with even $\operatorname{ord}_n(q)$ and Hermitian self-dual cyclic codes with odd $\operatorname{ord}_n(q)$. Furthermore, for Euclidean self-dual cyclic codes with odd $\operatorname{ord}_n(q)$ and Hermitian self-dual cyclic codes with even $\operatorname{ord}_n(q)$, we introduce a refined parameter selection that leads to larger minimum distances with the same code length and dimension. This approach also yields tighter lower bounds for several families of self-dual cyclic codes. This work enriches the theory of self-dual cyclic codes and offers new insights into estimating lower bounds on their minimum distances.

We demonstrate the application of Flow Matching, a technique originating from generative Artificial Intelligence, to probabilistic inversion in geophysical settings, such as seismic Full-Waveform inversion. We adapt the well-established mathematical theory of Flow Matching from generative Artificial Intelligence to the context of probabilistic inversion. We evaluate the approach with two case studies: a simple 2D velocity model to illustrate the general features of the method, and the OpenFWI dataset to show its capabilities for probabilistic inversion of more complex seismic velocity models.

We derive a heat kernel lower bound estimate for symmetric pure jump processes on general volume doubling metric measure spaces with possible degenerate and/or singular jump kernels using averaged jump kernels. As an application, the main result of this paper is applied to derive a lower bound estimate for the transition density function of the trace of Brownian motions on Sierpinski gaskets on the bottom of the Sierpinski gasket.

This paper investigates recovery operators in quasi-Nelson logic, the algebraizable logical counterpart of quasi-Nelson algebras. These form a variety of three-potent, distributive, but not necessarily involutive residuated lattices that may be regarded as a common generalization of Nelson and Heyting algebras. We consider both consistency and determinedness operators, with a particular focus on logics and algebras that satisfy the prelinearity condition, which is well-known in the area of mathematical fuzzy logics. We show that, essentially, all algebraic and logical results already proved for (prelinear, distributive) involutive residuated lattice-based LFIs/LFUs can be recovered in the quasi-Nelson setting, where one dispenses with the involutivity assumption. In this setting, consistency and undeterminedness operators are no longer duals of one another, and hence call for a more fine-grained algebraic and logical formalization.

Functional data analysis (FDA) provides statistical methods for analyzing samples of time-continuous stochastic processes. Measurements often arise in the form of sensor data for a key scientific variable. The practical problem of irregular sensor disruptions has fostered interest in analyzing partially observed random functions. Specifically, this paper is motivated by a time series of intermittently missing pollution data with dependence along pollution paths and missingness patterns. To allow statistical analysis, we develop the first inference methods for dependent, partially observed functional time series. Existing methods were not appropriate for this task, because they heavily rely on the independence of the data functions. Mathematically, we model data on the space of bounded functions equipped with the supremum norm. This allows simultaneous inference across the entire functional domain, including simultaneous confidence bands – something existing Hilbert-space-based methods cannot provide. To study non-stationary trends along the time series, we extend state-of-the-art multiscale inference methods (originally developed for scalar data) to partially observed functions. The key application of the latter methods is testing for excessive pollution levels in inner cities. Our approach combines state-of-the-art Gaussian approximations with stochastic process theory. Interestingly, it also improves existing results for fully observed functional time series by avoiding a functional CLT.

We consider the one-phase free boundary problem for the incompressible Navier-Stokes equations in $\mathbb{R}^d$ ($d\ge2$). The surface tension is taken into account. The initial domain, which is the outside of a bubble, is an exterior domain. We prove that there exists a bubble evolving by this free boundary problem which collapses in a finite time without blowing up of principal curvatures of its boundary. In other words, what is called a splash singularity is formed in a finite time. This type of result is also valid for a bounded initial domain. To construct such an example, we introduce the notion of a domain with $\delta$-wing which is a flat Riemannian manifold that is not embedded in $\mathbb{R}^d$, but it covers the $\delta$-neighborhood of the original domain whose boundary is self-intersected.

We study boundary conditions in GiMmsy's covariant and instantaneous formulations of classical field theories and show that the instantaneous state space in the presence of a constant Dirichlet boundary condition is a tangent bundle to the configuration space of fields satisfying said condition. We then study the instantaneous state space when only the velocity of the field is required to vanish at the boundary and show that this results in a sector structure, where each sector is a tangent bundle labeled by the configuration at the boundary. Taking the Legendre transform of this sectored state space yields a sectored phase space with leafwise canonical Poisson structures. We apply this to Yang-Mills theory with spatial boundary conditions and relate our results to flux superselection sectors. The sector-moving gauge transformations are not Hamiltonian because of the lack of a boundary momentum, prompting us to propose a novel definition of the asymptotic or boundary symmetry group as the quotient of the boundary-preserving Hamiltonian transformations by the trivial ones. The physical boundary symmetry group of electromagnetism is then shown to be a copy of the global gauge group even when all sectors are considered simultaneously. Conditions are discussed under which the same holds for non-Abelian Yang-Mills theory.

Andrews and Dastidar (Ramanujan J. 69, Article Number 26, (2026) ) introduced the $SOME(n)$ and $DSOME(n)$ functions that calculate the sum of all odd parts minus the sum of all even parts of ordinary partitions and distinct partitions, respectively of a positive integer $n$, and proved their generating functions and some congruences modulo 4 and 5. Recently, Gireesh and Hemanthkumar introduced an overpartition analogue of $SOME(n)$ function, denoted by $\overline{SOME}(n)$ and proved some congruences modulo 3, 5 and powers of 2. In this paper, we prove some new identities and congruences for $SOME(n)$, $DSOME(n)$, and $\overline{SOME}(n)$ functions, including monotonicity results. We also define a general analogue of $SOME(n)$ function, denoted by $S_{\mathcal P}(n)$, which calculates the sum of all odd parts minus the sum of all even parts in any arbitrary family of partitions $\mathcal P(n)$ of a positive integer $n$, and prove some divisibility properties. Additionally, we define a colour partition analogue of $SOME(n)$ function and prove divisibility properties.

For each prime $p$ and each power $q=p^k$, we present two large classes of permutation polynomials over $\F_{q^2}$ of the form $X^r B(X^{q-1})$ which have at most five terms, where $B(X)$ is a polynomial with coefficients in the prime field of $\F_{q^2}$ except at most one.

This paper establishes a mathematical framework for nonlinear subwavelength resonances and bound states in the continuum (BIC) in an acoustic metascreen with a cubic Kerr nonlinearity. We first use the quasiperiodic Dirichlet-to-Neumann operator to reduce the open resonance problem to an interior nonlinear variational problem. We then decompose the function space in which the variational problem is posed as the direct sum of two spaces and project the variational problem onto these two subspaces. Solving the projected equations successively yields a finite-dimensional nonlinear resonance equation with controlled remainders. We next apply the implicit function theorem near simple capacitance modes. This proves the existence and asymptotic expansions of linear subwavelength resonance branches and their small-amplitude nonlinear continuations. Finally, reflection symmetry gives a classification of the subwavelength branches. We characterize the symmetric resonance branches and prove that antisymmetric branches are exact BICs in both the linear problem and the nonlinear problem.

We construct closed manifolds with vanishing L^2-Betti numbers over every field) which do not virtually fibre over the circle. The class of fundamental groups that occurs is the largest possible, and in many cases the dimension may be taken to be six. We construct aspherical closed manifolds with residually (torsionfree and nilpotent) fundamental groups in all dimensions at least three whose L^2-Betti numbers vanish (over every field) and which do not virtually fibre over the circle. In particular this implies that in Kielak's Theorem about virtually algebraic fibring for RFRS-groups one cannot weaken the condition RFRS to residually (torsionfree and nilpotent.

We prove that the Rouquier dimension of the bounded derived category of finitely generated modules over a commutative noetherian ring is bounded below by the Krull dimension of the ring.

A total rainbow forest in an edge-colored graph is a forest that contains every edge color exactly once. Using a necessary and sufficient condition that a total rainbow forest exists, we demonstrate the existence of huge numbers of edge-colored graphs that are minimal obstructions to such existence.

We develop a rigorous numerical analysis framework for a class of semilinear parabolic problems with nonsmooth initial data. We employ a linear Galerkin finite element method for spatial discretization coupled with a high-order explicit exponential Runge-Kutta (EERK) temporal integration scheme. In contrast to conventional smooth error analysis, the nonsmooth case lacks a priori estimates for the higher-order derivatives of both the nonlinear term and the exact solution. By combining analytic semigroup techniques with fractional power space theory, we establish rigorous bounds for these derivatives. Finally, our analysis proves that the $p$th-order EERK method achieves a convergence rate of $\min(1 + \gamma/2 + \rho_1(\gamma)/2,\:p)$, where $\gamma$ characterizes the initial data regularity and $\rho_1(\gamma)$ quantifies the boundedness of the nonlinearity's first Fréchet derivative. Numerical experiments confirm the sharpness of these estimates.

We develop an analytic construction of nowhere-vanishing harmonic $1$-forms on real loci of K3-fibred Calabi-Yau $3$-folds with collapsing Ricci-flat Kähler metrics. We apply our construction to examples whose real loci have connected components diffeomorphic to $S^1\times S^2$ and to both trivial and nontrivial mapping tori. As an application, we produce examples of compact $7$-manifold with holonomy $G_2$ via the Joyce-Karigiannis construction.

We study the Diophantine equation $(a^n+1)(b^n+1)=x^2$, which belongs to the family of equations originating from the work of Szalay in 2000. If $a>1$, it is shown that the equation of the title has only one solution in positive integers, when $a$ and $b$ are distinct powers of the same integer $t>1$. Also, a complete description of the solutions is obtained under the assumptions that $a$ and $b$ are coprime and $n$ is even. Several other special cases of the equation are considered, and two conjectures are proposed.

The isometry group of the bounded Urysohn space, $G = \mathrm{Iso}(\U{1})$ is a central object in the study of Polish groups and topological dynamics. It is known that generic sequences in $G$ generate algebraically free dense subgroups. In this paper, we show that such generic free subgroups exhibit strong geometric rigidity. Specifically, we prove that for a comeager set of sequences generating dense free subgroups $F\leq G$, every non-trivial element $h\in F$ acts with maximal metric displacement, satisfying $\sup_{n\in \N} d(h^n(x),x) = 1$ for every $x \in \U{1}$. As a consequence, these generic subgroups satisfy the no small subgroup ($\nss$) property. We note that the method naturally extends to the full isometry group $\mathrm{Iso}(\mathbb{U})$ of the classical Urysohn space.

In this study, we develop a stochastic framework for computing Delta sensitivities in energy markets, where both prices and traded volumes are modeled as correlated stochastic processes. Within this framework, we analyze two complementary approaches for sensitivity analysis: the density method, which is applicable when the density of the underlying process is known, and the Malliavin calculus method, which does not require any explicit knowledge of the density and relies only on the dynamics of the processes. We present illustrative examples for both methods. For the density-based approach, we consider Ornstein-Uhlenbeck and CARMA processes to model prices and energy volumes. For the Malliavin calculus approach, we study Ornstein-Uhlenbeck processes, jump diffusion driven by a compound Poisson process, time-changed Brownian motion processes subordinated by an inverse Gaussian (IG) process, as well as Ornstein-Uhlenbeck processes driven by a normal inverse Gaussian (NIG) process. We provide some numerical examples illustrating the implementation of the proposed formulas and demonstrating a close agreement between the resulting delta estimates.

In this article, we investigate the local well-posedness of the nonlinear Schrödinger equation on the two-dimensional sphere $\mathbb{S}^2$: \begin{align*} i\partial_tu+\Delta_{g}u=F(u). \end{align*} The nonlinearity $F(u)$ is assumed to be gauge-invariant. More presicely, there exists a function $V\in C^\infty(\mathbb{C},\mathbb{R})$ such that $F=\frac{\partial V}{\partial \bar{z}}$. Moreover, $V(z)$ obeys \begin{equation}\label{H-11} V(e^{i\theta}z)=V(z),\,\,\theta\in\Bbb R,\,\,z\in\Bbb C, |\partial_z^{k_1}\partial_{\bar{z}}^{k_2}V(z)|\leq C_{k_1,k_2}(1+|z|)^{1+\alpha-k_1-k_2},\tag{H-1} \end{equation} for some $\alpha\geq3.$ The main contribution of this paper is the new lower bound of threshold of local well-posedness $s_c(\mathbb{S}^2,\alpha)$. Specifically, under assumption \eqref{H-11}, we prove that for $\alpha \geq 3$, the equation is ill-posed in $H^s(\mathbb{S}^2)$ with $s < 1 - \frac{2}{\alpha-1}$ in the sense that the norm inflation occurs. Combined with the well-posedness in Yang [Sci. China Math. 58 (2015), 1023-1046], the exact threshold $s_c(\mathbb{S}^2,\alpha)$ for $\alpha\geq5$ is $1-\frac{2}{\alpha-1}$, which matches the scaling-critical regularity as the Euclidean setting. Moreover, for $\alpha \in [3, \frac{11}{3})$, we show that the solution map is not uniformly continuous in the range $0 < s < \frac14$ for the power-type nonlinearity $F(u)=|u|^{\alpha-1}u$, which lies strictly above the scaling-invariant threshold. This provides a new characterization of the ill-posedness regime for all $\alpha \geq 3$, extending an earlier result of Burq-Gérard-Tzvetkov [Math. Res. Lett. 9 (2002), 323-335]. Our result can also be regarded as a Schrödinger counterpart of Xia [Int. Math. Res. Not. (2021), 15533-15554].

We introduce a new cumulant-based quantum relative Rényi functional as a candidate quantum Rényi divergence, derived from the cumulant-generating function (CGF) of the quantum relative surprisal operator and extending the classical connection between Rényi divergence and statistical cumulants to the quantum setting. Unlike the Petz and sandwiched quantum Rényi divergences, the proposed construction is motivated by statistical structure rather than operator-algebraic or operational principles. The functional naturally admits a path-integral-like representation through the Lie-Trotter product expansion, providing a trajectory-based interpretation of quantum divergence in Hilbert space. On its natural non-regularized domain for $\alpha>1$ under the support condition $\operatorname{supp}(\rho)\subseteq\operatorname{supp}(\sigma)$, we establish several fundamental properties, including positivity, reduction to the classical case, additivity, unitary invariance, continuity, and monotonicity with respect to the Renyi parameter $\alpha$. Whether the functional satisfies the quantum data-processing inequality (QDPI) under arbitrary CPTP maps remains open. To extend the analysis beyond the studied regime, we introduce a regularized version of the functional and study its behavior at $\alpha=0$. We show that the resulting relative quantumness quantity vanishes if and only if the underlying states commute, yielding a necessary and sufficient characterization of non-commutativity. For commutativity-preserving (CoP) channels, we further conjecture a QDPI-type monotonicity relation for this quantity. Extensive numerical simulations provide strong evidence in support of this conjecture, with no violations observed for the CoP channels considered in this work.

We show that there is no infinite spin at total collisions for $-\kappa$-homogeneous N-body problem in higher dimensional Euclidean space $\mathbb{R}^d$, in which $0 < \kappa < 2$ ($\kappa = 1$ the Newtonian case), provided the limiting normalized central configuration is isolated and is of dimension d or d - 1. In the Newtonian case $\kappa = 1$, this extends the work of Moeckel-Montgomery to $d \ge 3$ and in the d = 3 case offers a different approach as compared to the current preprint of Pinzari-Zgliczynski.