A right $A$-module $M$ is said to be generalized bassian if the existence of an injective homomorphism $M\to M/N$ for some submodule $N$ of $M$ implies that $N$ is a direct summand of $M$. We describe singular generalized bassian modules over non-primitive Dedekind prime rings.\\ The study is supported by grant of Russian Science Foundation.

This paper investigates the multi-compartment vehicle routing problem with multiple time windows (MCVRPMTW), an extension of the classical vehicle routing problem with time windows that considers vehicles equipped with multiple compartments and customers requiring service across several delivery time windows. The problem incorporates three key compartment-related features: (i) compartment flexibility in the number of compartments, (ii) item-to-compartment compatibility, and (iii) item-to-item compatibility. The problem also accommodates practical operational requirements such as driver breaks. To solve the MCVRPMTW, we develop an exact branch-and-price (B&P) algorithm in which the pricing problem is solved using a labeling algorithm. Several acceleration strategies are introduced to limit symmetry during label extensions, improve the stability of dual solutions in column generation, and enhance the branching process. To handle large-scale instances, we propose a rolling-space B&P algorithm that integrates clustering techniques into the solution framework. Extensive computational experiments on instances inspired by a real-world industrial application demonstrate the effectiveness of the proposed approach and provide useful managerial insights for practical implementation.

We develop a unified density-based framework for primality, coprimality, and prime pairs, and introduce an intrinsic normalized model for prime gaps constrained by the Prime Number Theorem. Within this setting, a structural tension between Hardy-Littlewood, Cramer, and PNT predictions emerges, leading to quantitative estimates on the rarity of extreme gaps. Additive representations of even integers are reformulated as local density problems, yielding non-conjectural upper and lower bounds compatible with Hardy-Littlewood heuristics. Finally, the Riemann zeta function is analyzed via truncated Euler products, whose stability and oscillatory structure provide a coherent interpretation of the critical line and prime-based numerical criteria for the localization of non-trivial zeros.

In this paper, we study the norm-controlled inversion problem in two classes of algebras of integrable functions. In contrast of the classical case of $L^{1}(G)$, we prove that this problem has a positive solution in our setting without any additional restrictions.

In this paper, we prove a Pohozaev identity for the Spectral Fractional Laplacian (SFL). This identity allows us to establish non-existence results for the semilinear Dirichlet problem $(-Δ|_Ω)^su = f(u)$ in star-shaped domains. The first such identity for non-local operators was established by Ros-Oton and Serra in 2014 for the Restricted Fractional Laplacian (RFL). However, the SFL differs fundamentally from the RFL, and the integration by parts strategy of Ros-Oton and Serra cannot be applied. Instead, we develop a novel spectral approach that exploits the underlying quadratic structure. Our main result expresses the identity as a Schur product of the classical Pohozaev quadratic form and a transition matrix that depends on the eigenvalues of the Laplacian and the fractional exponent.

In this paper we study higher even Gaussian maps of the canonical bundle for cyclic trigonal curves. More precisely, we study suitable restrictions of these maps determining a lower bound for the rank, and more generally, a lower bound for the rank for the general trigonal curve. We also manage to give the explicit description of the kernel of the second Gaussian map. Finally, we use these results to show the non existence of "extra" asymptotic directions for cyclic trigonal curves in some spaces generated by higher Schiffer variations.

We study a system of Forward-Backward Stochastic Differential Equations (FBSDEs) with time-delayed generators. The forward process includes a reflection component expressed via a Stieltjes integral, while the backward process takes the form of a Generalized BSDE. We establish the connection between this FBSDE system and non-linear path-dependent PDEs with Neumann boundary conditions by deriving a representation formula.

Given a Riemannian manifold $M$ endowed with a smooth metric $g$ satisfying upper and lower sectional curvature bounds, we show an equivalence property between the $\mathrm{L}^2$ norm on $M$ and the $\mathrm{L}^2$ norm on subsets $ω$ satisfying a thickness condition, for functions in the range of a spectral projector. The thickness condition is known to be optimal in this setting. The constant appearing in the equivalence of norms property depends only on the dimension of the manifold, curvature bounds, and frequency threshold of the spectral cutoff, but, crucially, not on the injectivity radius.

In 1980, Odoni initiated the study of the fixed-point proportion of iterated Galois groups of polynomials motivated by prime density problems in arithmetic dynamics. The main goal of the present paper is to completely settle the longstanding open problem of computing the fixed-point proportion of geometric iterated Galois groups of polynomials. Indeed, we confirm the well-known conjecture that Chebyshev polynomials are the only complex polynomials whose geometric iterated Galois groups have positive fixed-point proportion. Our proof relies on methods from group theory, ergodic theory, martingale theory and complex dynamics. This result has direct applications to the proportion of periodic points of polynomials over finite fields. The general framework developed in this paper applies more generally to rational functions over arbitrary fields and generalizes, via a unified approach, previous partial results, which have all been proved with very different methods.

Define the codewords of the Tensor Reed-Muller code $\mathsf{TRM}(r_1,m_1;r_2,m_2;\dots;r_t,m_t)$ to be the evaluation vectors of all multivariate polynomials in the variables $\left\{x_{ij}\right\}_{i=1,\dots,t}^{j=1,\dots m_i}$ with degree at most $r_i$ in the variables $x_{i1},x_{i2},\dots,x_{im_i}$. The generator matrix of $\mathsf{TRM}(r_1,m_1;\dots;r_t,m_t)$ is thus the tensor product of the generator matrices of the Reed-Muller codes $\mathsf{RM}(r_1,m_1),\dots, \mathsf{RM}(r_t,m_t)$. We show that for any constant rate $R$ below capacity, one can construct a Tensor Reed-Muller code $\mathsf{TRM}(r_1,m_1;\dotsc;r_t,m_t)$ of rate $R$ that is decodable in quasilinear time. For any blocklength $n$, we provide two constructions of such codes: 1) Our first construction (with $t=3$) has error probability $n^{-ω(\log n)}$ and decoding time $O(n\log\log n)$. 2) Our second construction, for any $t\geq 4$, has error probability $2^{-n^{\frac{1}{2}-\frac{1}{2(t-2)}-o(1)}}$ and decoding time $O(n\log n)$. One of our main tools is a polynomial-time algorithm for decoding an arbitrary tensor code $C=C_1\otimes\dotsc\otimes C_t$ from $\frac{d_{\min}(C)}{2\max\{d_{\min}(C_1),\dotsc,d_{\min}(C_t) \}}-1$ adversarial errors. Crucially, this algorithm does not require the codes $C_1,\dotsc,C_t$ to themselves be decodable in polynomial time.

In this paper we present a generalization of the Fox H-function called Fox-Barnes J-function. Like the Fox H-function, it is defined as a contour integral in the complex plane, but instead of an integrand given by a ratio of products of gamma functions involving several parameters, we use a ratio of products of double gamma functions. We study the conditions for its existence and how to choose a contour of integration based on the involved parameters. We discuss how the Fox H-function appears as a particular case and prove some properties of the Fox-Barnes J-function. As an application, we show how the Laplace transform of the Kilbas-Saigo function can be conveniently written in terms of the Fox-Barnes J-function, even in cases where the usual series representation is not convergent.

We consider the following task: how for a given quantum state $ρ$ to find a grounded Hamiltonian $H$ satisfying the condition $\mathrm{Tr} Hρ\leq E_0<+\infty$ in such a way that the von Neumann entropy of the Gibbs state $γ_H(E)$ corresponding to a given energy $E>0$ be as small as possible. We show that for any mixed state $ρ$ with finite entropy and any $E>0$ there exists a solution $H(ρ,E_0,E)$ of the above problem (unique in the non-degenerate case) which we call optimal Hamiltonian for the state $ρ$. Explicit expressions for $H(ρ,E_0,E)$, $γ_{H(ρ,E_0,E)}(E)$ and $S(γ_{H(ρ,E_0,E)}(E))$ are obtained. Analytical properties of the function $E\mapsto S(γ_{H(ρ,E_0,E)}(E))$ are explored. Several examples are considered. We also consider a modification of the above task in which arbitrary Hamiltonians (not necessarily grounded) are considered. The basic application motivated this research is described. As examples, new semicontinuity bounds for the von Neumann entropy and for the entanglement of formation are obtained and briefly discussed (with the intention to give a detailed analysis in a separate article).

Magnetic Particle Imaging (MPI) is a promising tomographic technique for visualizing the spatio-temporal distribution of superparamagnetic nanoparticles, with applications ranging from cancer detection to real-time cardiovascular monitoring. Traditional MPI reconstruction relies on either time-consuming calibration (measured system matrix) or model-based simulation of the forward operator. Recent developments have shown the applicability of Chebyshev polynomials to multi-dimensional Lissajous Field-Free Point (FFP) scans. This method is bound to the particular choice of sinusoidal scanning trajectories. In this paper, we present the first reconstruction on real 2D MPI data with a trajectory-independent model-based MPI reconstruction algorithm. We further develop the zero-shot Plug-and-Play (PnP) algorithm of the authors -- with automatic noise level estimation -- to address the present deconvolution problem, leveraging a state-of-the-art denoiser trained on natural images without retraining on MPI-specific data. We evaluate our method on the publicly available 2D FFP MPI dataset ``MPIdata: Equilibrium Model with Anisotropy", featuring scans of six phantoms acquired using a Bruker preclinical scanner. Moreover, we show reconstruction performed on custom data on a 2D scanner with additional high-frequency excitation field and partial data. Our results demonstrate strong reconstruction capabilities across different scanning scenarios -- setting a precedent for general-purpose, flexible model-based MPI reconstruction.

The integration of Terrestrial and Non-Terrestrial Networks (TN-NTNs), introduced in 5G, is advancing toward a unified and seamless network of networks in Sixth-Generation (6G). This evolution markedly increases the volume of generated and exchanged data, imposing stringent technical and operational requirements along with higher cost and energy consumption. Consequently, efficient management of data generation and transmission within this unified architecture has become essential. In this article, we investigate semantics-aware information handling in unified TN-NTNs, where data communication between distant TN nodes is enabled via an NTN. We consider an Internet of Things (IoT) monitoring system in which status updates from a remote Energy Harvesting (EH) device are delivered to a destination monitor through a network of Low Earth Orbit (LEO) satellites. We leverage semantic metrics, such as Query Version Age of Information, which collectively capture the timeliness, relevance, and utility of information. This approach minimizes the transmission of stale, uninformative, or unusable information, thereby reducing the volume of data that must be transmitted and processed. The result is a substantial reduction in energy consumption and data exchange within the network-achieving up to 73% lower energy-charging requirements and fewer transmission demands than the state of the art-without compromising the conveyed information.

We introduce a general technique to construct tight fusion frames with prescribed symmetries. Applying this technique with a prescription for "all the symmetries", we construct a new family of equi-isoclinic tight fusion frames (EITFFs), which consequently form optimal Grassmannian codes. By virtue of their construction, our EITFFs have the remarkable property of total symmetry: any permutation of subspaces can be achieved by an appropriate unitary.

We introduce a unified theory of Cartan subgroups and maximal toroids - defined as connected multiplicative type subgroups that are maximal amongst all such subgroups - which holds for all affine algebraic groups over a field, regardless of smoothness. For instance we show that maximal toroids always exist, that they are invariant under base change, and that they are in natural 1-1 correspondence with Cartan subgroups. Our results generalise known results for Cartan subgroups and maximal tori of smooth affine algebraic groups, as well as their analogues for restricted Lie algebras. We conclude with some applications to, and a brief discussion of, some generation problems for algebraic groups.

We present a method for associating labeled directed graphs to finite-dimensional Lie algebras, thereby enabling rapid identification of key structural algebraic features. To formalize this approach, we introduce the concept of graph-admissible Lie algebras and analyze properties of valid graphs given the antisymmetry property of the Lie bracket as well as the Jacobi identity. Based on these foundations, we develop graph-theoretic criteria for solvability, nilpotency, presence of ideals, simplicity, semisimplicity, and reductiveness of an algebra. Practical algorithms are provided for constructing such graphs and those associated with the lower central series and derived series via an iterative pruning procedure. This visual framework allows for an intuitive understanding of Lie algebraic structures that goes beyond purely visual advantages, since it enables a simpler and swifter grasping of the algebras of interest beyond computational-heavy approaches. Examples, which include the Schrödinger and Lorentz algebra, illustrate the applicability of these tools to physically relevant cases. We further explore applications in physics, where the method facilitates computation of similtude relations essential for determining quantum mechanical time evolution via the Lie algebraic factorization method. Extensions to graded Lie algebras and related conjectures are discussed. Our approach bridges algebraic and combinatorial perspectives, offering both theoretical insights and computational tools into this area of mathematical physics.

In this paper, we address the model reduction problem for linear hybrid systems via the interconnection-based technique called moment matching. We consider two classical interconnections, namely the direct and swapped interconnections, in the hybrid setting, and we present families of reduced-order models for each interconnection via a hybrid characterisation of the steady-state responses. By combining the results for each interconnection, the design of a reduced-order model that achieves moment matching simultaneously for both interconnections is studied. In addition, we show that the presented results have simplified counterparts when the jumps of the hybrid system are periodic. A numerical simulation is finally given to illustrate the results.

This paper presents a universal numerical scheme tailored for tackling linear integral, integro-differential, and both initial and boundary value problems of ordinary differential equations. The numerical scheme is readily adapted for resolving ill-posed problems. Central to our approach is high-degree piecewise-polynomial approximation to the exact solution. We illustrate the accuracy and stability of our numerical solutions in the presence of noise through illustrative examples. Additionally, we demonstrate that proposed regularization being applied to high-degree interpolation, effectively eliminates Runge's phenomenon.

We show the failure of the pointwise convergence of averages along the Omega function in a number field. As a consequence, we show, for instance, that the averages \[ \frac{1}{N^2}\sum_{1\leq m,n \leq N} f(T^{Ω(m^2+n^2)}x)\] do not converge pointwise in ergodic systems, addressing a question posed by Le, Moreira, Sun, and the second author. On the other hand, using number-theoretic methods, we establish the pointwise convergence of averages along the $Ω$ function defined on the ideals of a number field in uniquely ergodic systems. Using this dynamical framework, we also derive several natural number-theoretic consequences of independent interest.

Let $F$ be a non-archimedean local field of characteristic different from $2$ and of residual characteristic $p$. We generalise the theory of the Weil representation over $F$ with complex coefficients to $\ell$-modular representations \textit{i.e.} when the complex coefficients are replaced by a coefficient field $R$ of characteristic $\ell \neq p$. We obtain along the way a generalisation of the Stone-von Neumann theorem to the $\ell$-modular setting, together with the Weil representation with coefficients in $R$ on the $R$-metaplectic group. Surprisingly enough, the latter $R$-metaplectic group happens to be split over the symplectic group if $\ell = 2$. The theory also makes sense when $F$ is a finite field of odd characteristic. We also establish the irreducibility of the theta lift in the cuspidal case as long as $\ell$ does not divide the pro-orders of the groups at stake and we provide a compatibility to congruences in this setting via an integral version of the theta lift.

We introduce a class of pseudo-bosonic Klein-Gordon fields in 1+1 dimensions and we discuss some of their properties. This work originates from non Hermitian quantum mechanics and deformed canonical commutation relations. We show that, within this class of fields, there exist a specific subclass with the interesting feature of having finite equal space-time two-points function, contrarily to what happens for {\em standard} Klein-Gordon fields. This, in our opinion, is a relevant aspect of our proposal which is a good motivation to undertake a deeper analysis of this (and related) quantum fields.

We study the proximity operator of the nonconvex, scale-invariant ratio $h(\vx)=\|\vx\|_{1}/\|\vx\|_{2}$ and show it can be computed exactly in any dimension. By expressing $\vx=r\vu$ and exploiting sign and permutation invariance, we reduce the proximal step to a smooth optimization of a rank-one quadratic over the nonnegative orthant of the unit sphere. We prove that every proximal point arises from a finite candidate set indexed by $k\in\{1,\dots,n\}$: the active subvector is a local, but nonglobal, minimizer on $\mathbb{S}^{k-1}$ characterized by the roots of an explicit quartic. This yields closed-form candidates, an exact selection rule, and a necessary and sufficient existence test. Building on these characterizations, we develop practical algorithms, including an $O(n)$ implementation via prefix sums and a pruning criterion that avoids unnecessary quartic solves. The method returns all proximal points when the prox is non-unique, and in experiments it attains strictly lower objective values than approaches that guess sparsity or rely on sphere projections with limited scalability.

In this paper, we consider the problem of reconstructing piece-wise smooth functions from their non-uniform Fourier data. We first extend the filter method for uniform Fourier data to the non-uniform setting by using the techniques of admissible frames. We show that the proposed non-uniform filter method converges exponentially away from the jump discontinuities. However, the convergence rate is significantly slower near the jump discontinuities due to the Gibbs phenomenon. To overcome this issue, we combine the non-uniform filter method with a stable extrapolation method to recover the function values near the jump discontinuities. We show that the proposed hybrid method could achieve exponential accuracy uniformly on the entire domain. Numerical experiments are provided to demonstrate the performance of the proposed method.

In this contribution we propose an exponential update algorithm for magnetic moments appearing in the framework of micromagnetics and the Landau-Lifshitz-Gilbert (LLG) equation. This algorithm can be interpreted as the geometric integration on spheres, that a priori satisfy the unit length constraint of the normalized magnetization vector. Even though the geometric structures for this are obvious and some works already use an exponential algorithm, to the best of the authors' knowledge, there is no canonical structure of the LLG equation for the exponential update algorithm in micromagnetism. Tensor algebraic reformulations of the LLG equation allow the canonical representation of the evolution equation for the magnetization, which serves as the basis for different integrators. Based on the specific structure of the exponential of skew symmetric matrices an efficient update scheme is derived. The excellent performance of the proposed exponential update algorithm is demonstrated in representative examples.

McDonald and Clerk [Phys.\ Rev.\ Research 5, 033107 (2023)] showed that for linear open quantum systems the Liouvillian spectrum is independent of the noise strength. We first make this noise-independence principle precise in continuous time for multimode bosonic Gaussian Markov semigroups: for Hurwitz drift, a time-independent Gaussian similarity fixed by the Lyapunov equation gauges away diffusion for all times, so eigenvalues and non-diagonalizability are controlled entirely by the drift, while diffusion determines steady states and the structure of eigenoperators. We then extend the same separation to discrete time for general stable multimode bosonic Gaussian channels: for any stable Gaussian channel, we construct an explicit Gaussian similarity transformation that gauges away diffusion at the level of the channel parametrization. We illustrate the method with a single-mode squeezed-reservoir Lindbladian and with a non-Markovian family of single-mode Gaussian channels, where the exceptional-point manifolds and the associated gauging covariances can be obtained analytically.

For closed manifolds with compact Lie group actions, we study Austin-Braam's Morse-theoretic construction of Borel equivariant cohomology using the technique of stabilization. We show that a $C^1$-small equivariant perturbation produces stable invariant Morse-Bott functions. This allows us to realize the equivariant transversality and orientability assumptions in Austin-Braam's framework by choosing generic invariant Riemannian metrics.

A rich $k$-flow is a nowhere-zero $k$-flow $ϕ$ such that, for every pair of adjacent edges $e$ and $f$, $|ϕ(e)| \neq |ϕ(f)|$. A graph is rich flow admissible if it admits a rich $k$-flow for some integer $k$. In this paper, we prove that if $G$ is a rich flow admissible graph with maximum degree $Δ$, then $G$ admits a rich $(264Δ- 445)$-flow.

In this paper, we study the spherical cardioid distribution, a higher-dimensional and higher-order generalization of the circular cardioid distribution. This distribution is rotationally symmetric and generates unimodal, multimodal, axial, and girdle-like densities. We show several characteristics of the spherical cardioid that make it highly tractable: simple density evaluation, closedness under convolution, explicit expressions for vectorized moments, and efficient simulation. The moments of the spherical cardioid up to a given order coincide with those of the uniform distribution on the sphere, highlighting its closeness to the latter. We derive estimators by the method of moments and maximum likelihood, their asymptotic distributions, and their asymptotic relative efficiencies. We give the machinery for a bootstrap goodness-of-fit test based on the projected-ecdf approach, including the projected distribution and closed-form expressions for test statistics. An application to modeling the orbits of long-period comets shows the usefulness of the spherical cardioid distribution in real data analyses.

We study the behavior of birational automorphism groups in families of projective hyper-Kähler manifolds.