This paper develops a rich theory of cardinality in the paraconsistent and paracomplete set theory $\mathrm{BZFC}$, where sets can be inconsistent ($A$ such that ``$x\in A$'' is both true and false for some $x$) or incomplete ($A$ such that ``$x\in A$'' is neither true nor false for some $x$). We carefully analyze what it means for two potentially incomplete or inconsistent sets to have ``the same size'', construct the corresponding cardinal numbers, and develop the basic theory of cardinal arithmetic. A surprising result is that the cardinality of any set can be expressed as a linear combination of three fundamental cardinal numbers with classical cardinals as coefficients. In that sense, our cardinal numbers form a three-dimensional space over the usual cardinals, much like how the complex numbers form a two-dimensional space over the reals.

Homeomorphisms of the Cantor set play a central role in topology, dynamical systems and descriptive set theory. In parallel, several classes of fence-like spaces - such as the hairy Cantor set, hairy arcs, Cantor bouquets in complex dynamics, the Lelek fan in topology and Fraïssé fence in descriptive set theory - have recently been studied for their rich structural and dynamical properties. In this paper, we introduce a general construction that associates to each homeomorphism of the Cantor set a canonically defined homeomorphism of a corresponding fence space. This construction lifts dynamical properties from the Cantor set to these fence-like spaces, allowing one to systematically transfer features such as minimality, recurrence, and orbit structure. As a consequence, we obtain a unified framework for studying dynamics on a broad class of fence-like spaces and establish new connections between their topological structure and induced dynamical behavior.

Horizontal stirred bed reactors (HSBRs) are used in gas--phase polyolefin production, where efficient solids mixing and controlled residence time distributions are essential for product quality and stability. Despite their industrial relevance, the influence of operating conditions on granular flow and mixing in HSBRs is not well understood. Discrete Element Method (DEM) simulations are used to study the effects of rotation speed and fill level on particle motion, mixing, and axial transport in a lab--scale HSBR. An industrial--grade polypropylene powder is modelled using calibrated contact parameters. Mixing is quantified using the Lacey index in axial (z) and cross--sectional (xy) directions. Particle circulation is characterised via cycle--time analysis and a coarse--grained angular velocity field. Axial dispersion coefficients are obtained from particle trajectories using both Einstein--type and cycle--based approaches, and validated with a diffusion model predicting the axial Lacey index. Results show that axial mixing depends strongly on rotation speed and fill level: higher rotation speeds accelerate homogenization, while higher fill levels slow mixing. Cross--sectional mixing is mainly sensitive to rotation speed, with fill--level effects diminishing at higher speeds. Cycle time decreases with increasing rotation speed and fill level, indicating enhanced circulation. Axial dispersion increases with rotation speed but decreases with fill level, with consistent results across methods. These findings reveal trade--offs between axial mixing, circulation, and dispersion, highlighting the need to balance operating conditions and demonstrating the capability of DEM to support HSBR optimisation.

This paper presents an indirect data-driven output feedback controller synthesis for nonlinear systems, leveraging Structured State-space Models (SSMs) as surrogate models. SSMs have emerged as a compelling alternative in modelling time-series data and dynamical systems. They can capture long-term dependencies while maintaining linear computational complexity with respect to the sequence length, in comparison to the quadratic complexity of Transformer-based architectures. The contributions of this work are threefold. We provide the first analysis of controllability and observability of SSMs, which leads to scalable control design via Linear Matrix Inequalities (LMIs) that leverage contraction theory. Moreover, a separation principle for SSMs is established, enabling the independent design of observers and state-feedback controllers while preserving the exponential stability of the closed-loop system. The effectiveness of the proposed framework is demonstrated through a numerical example, showcasing nonlinear system identification and the synthesis of an output feedback controller.

We classify $n\times n$-matrix-valued continuous commutativity and spectrum preservers defined on spaces of (a) normal, (b) semisimple and (c) arbitrary $n\times n$ matrices with spectra contained in sufficiently connected subsets $\mathcal{X}\subseteq \mathbb{C}$, generalizing a number of results due to Šemrl, Gogić, Tomašević and the author among others. In case (a) these are always conjugations or transpose conjugations, while in cases (b) and (c) qualitatively distinct possibilities arise depending on the local regularity of the complex-conjugation map close to coincident-eigenvalue loci of $\mathcal{X}^n$.

Given a connected reductive algebraic group $G$, we consider the class of spherical subgroups $H \subset G$ such that $H$ is regularly embedded in a parabolic subgroup $P \subset G$ and $H,P$ have a common Levi subgroup $L$. In a previous paper, the author developed a fast algorithm that reduces the computation of the set of spherical roots for such subgroups $H$ to the case where the quotient of Lie algebras $\operatorname{Lie} P / \operatorname{Lie} H$ is a strictly indecomposable spherical $L$-module. In this paper, we complete the classification of all such cases and compute the spherical roots for each of them, which enables one to use the above fast algorithm directly for computing the spherical roots for arbitrary spherical subgroups in the class under consideration.

We establish the Hasse principle for $100\%$ of conic bundles over $\mathbb{P}^1_{\mathbb{Q}}$.

We construct a causal and covariantly stable kinetic model whose spectrum at real wavenumbers $k$ reproduces any rest-frame stable dissipative dispersion relation $ω(k)$ via suitable initialization of the microscopic degrees of freedom. Macroscopic observables can therefore obey arbitrary linear evolution equations (including forms that would be acausal if taken as fundamental), while the underlying dynamics remains causal, and all apparent propagation is encoded in the initial data. This provides an explicit counterexample to the idea that microscopic causality alone constrains the analytic form of dispersion relations at real $k$. In particular, bounds on transport coefficients based solely on the analytic structure of $ω(k)$, such as the hydrohedron bounds, require additional assumptions about the region in the complex $k$-plane where $ω(k)$ corresponds to physical modes.

This article presents a novel and comprehensive approach for analyzing bending behavior of the tapered perforated beam under an exponential load. The governing differential equation includes important factors like filling ratio ($α$), number of rows of holes ($N$), tapering parameters ($ϕ$ and $ψ$), and exponential loading parameter ($γ$), providing a realistic and flexible representation of perforated beam configuration. Main goal of this work is to see how well the Domain mapped physics-informed Functional link Theory of Functional Connection (DFL-TFC) method analyses bending response of perforated beam with square holes under exponential loading. For comparison purposes, a corresponding PINN-based formulation is developed. Outcomes clearly show that the proposed DFL-TFC framework gives better results, including faster convergence, reduced computational cost, and improved solution accuracy when compared to the PINN approach. These findings highlight effectiveness and potential of DFL-TFC method for solving complex engineering problems governed by differential equations. Within this framework, hidden layer is replaced by a functional expansion block that enriches input representation via orthogonal polynomial basis functions, and the domain of DE mapped to corresponding domain of orthogonal polynomials. A Constrained Expression (CE), constructed through the Theory of Functional Connections (TFC) using boundary conditions, ensures that constraints are exactly satisfied. In CE, free function is represented using a Functional Link Neural Network (FLNN), which learns to solve resulting unconstrained optimization problem. The obtained results are further validated through the Galerkin and PINN solutions.

We study set-valued decision rules in which performance is defined by the inclusion of the top-$p$ hypotheses, rather than only the single best or true hypothesis. This criterion is motivated by sensor selection for target tracking, where inexpensive measurements are used to identify a list of sensor nodes that are likely to be closest to a target. We analyze the performance of top-$p$ versus top-$1$ selection under sequential hypothesis testing, propose a geometry-aware sensor selection algorithm, and validate the approach using real testbed data.

The concept of freely falling frames suggests that gravity exhibits a local Lorentz gauge symmetry and requires a background Minkowski reference frame. The gauge vector fields of a Yang-Mills-type theory can be constructed from the transformations to these local freely falling frames, thereby leading to a composite theory of gravity. We propose coordinate conditions under which an exact black-hole solution can be obtained. Our analysis of planar gravitational waves reveals that, despite the large symmetry group, composite gravity possesses only four physical degrees of freedom. We elaborate a Hamiltonian formulation of composite gravity, derive the full set of constraints for the nonlinear theory, and outline the pathway toward its quantization.

We study an overdetermined elliptic free boundary problem for exterior domains in $\mathbb{R}^N$, $N \ge 2$, introduced by F. Morabito [Comm. PDE 46 (2021), 1137-1161]. The overdetermining condition prescribes the Neumann data as a multiple of the boundary mean curvature, with parameter $Γ$, together with a spherical compatibility condition. For $N \ge 3$, we prove rigidity of the spherical solution among star-shaped domains when $Γ\ge N-2$; in the borderline case $Γ= N-2$, the star-shapedness assumption can be removed, and rigidity holds among all bounded domains. The proof combines the Pohozaev identity, geometric identities, and the sharp boundary inequality of Agostiniani and Mazzieri for capacitary potentials. We also obtain rigidity among bounded domains for $Γ\le 0$ via Serrin's moving plane method. In dimension two, the unit disc is the only admissible domain for every $Γ$.

We prove that $\mathrm{SL}_3(\mathbb{Z})$ contains a non-split sharply 2-transitive subgroup, answering a question of Glasner and Gulko. We also prove that $\mathrm{SL}_4(\mathbb{Z})$ contains a non-split sharply 3-transitive subgroup, but that $\mathrm{SL}_3(\mathbb{Z})$ does not contain an infinite sharply 3-transitive subgroup.

A graph $G$ is $k$-vertex-critical if $χ(G)=k$ but $χ(G-v)<k$ for all $v\in V(G)$. In this paper we make progress on the open problem of the finiteness of $k$-vertex-critical $(P_4+\ell P_1)$-free graphs by showing that there are only finitely many $k$-vertex-critical graphs in the following subfamilies of $(P_4+\ell P_1)$-free graphs for all $k\ge 1$ and $\ell\ge 0$: $\bullet$ $(P_4+\ell P_1,\text{chair})$-free graphs, $\bullet$ $(P_4+\ell P_1,P_5,\text{bull})$-free graphs, and $\bullet$ $(P_4+\ell P_1,P_5,\text{cricket})$-free graphs. In fact, all but the first of these are special cases of our general result that there are only finitely many $k$-vertex-critical $(P_4+\ell P_1,B_{4}(m),B_{3}(m)^{+})$-free graphs for all $k\ge 1$ and $\ell,m\ge 0$. Here $B_{n}(m)$ is the graph obtained from a path of order $n$ by identifying one of its leaves with the centre vertex of $K_{1,m}$ and $B_{n}(m)^{+}$ is the graph obtained by identifying an edge of $K_3$ with the edge of $B_{n}(m)$ with endpoints of degrees $2$ and $m$, respectively. Our results imply the existence of simple polynomial-time certifying algorithms to decide the $k$-colourability of all graphs in these subfamilies for every fixed $k$. We also show that $χ(G)\le \ell+2$ for all $(P_4+\ell P_1,K_3)$-free graphs and all $\ell\ge 0$, improving the previously known upper bound of $2\ell+2$ that followed from Randerath and Schiermeyer's 2004 result on $(P_t,K_3)$-free graphs. More generally, we provide a $χ$-bound in $O(\ell^{ω-1})$ for $(P_4+\ell P_1)$-free graphs which improves the bound of $(2\ell+2)^{ω-1}$ which followed from Gravier, Hoàng and Maffray in 2003 for $P_{t}$-free graphs.

It is proved that the C*-algebra of a graph is residually finite dimensional (RFD) if and only if the graph has no infinite receiver, no cycle with an exit, no infinite ackward chain and from each vertex, there is a finite path to a sink or a cycle or an infinite emitter.

We show that the global unique continuation principle holds for the parabolic fractional $p-$Laplace equation with very rough potentials $V(x,t) \in L^{p'}_tW^{-s,p'}_x$. Whereas the result is new even for the fractional $p-$Laplace operator, the corresponding local problem remains open even with zero potential. The short proof eschews extension techniques and Carleman estimates.

A real Lie algebra defines by extension of scalars a complex Lie algebra that is isomorphic to its Galois conjugate. In this paper, we are interested in the converse property: given a complex Lie algebra that is isomorphic to its conjugate, is it defined over the real numbers? We prove the existence of a $10$-dimensional nilpotent complex Lie algebra for which the answer is negative, disproving a recent conjecture by Deré. In addition, we compute the generic obstruction to this descent problem in terms of Brauer groups.

This lecture note addresses the common misconception that the Gaussian distribution always yields the largest Cramér-Rao Bound (CRB). We show that this property only holds under restrictive conditions: specifically, when the mean and covariance parameters are decoupled in the Fisher Information Matrix (FIM), when the parameter of interest lies in the mean vector and when there are no additive nuisance parameters. Beyond this framework, we provide counterexamples demonstrating that non-Gaussian distributions can produce larger CRB.

We give a necessary and sufficient condition on a matrix for its centralizer in $\sf{GL}(n,\mathbb{Z})$ to be polycyclic, or equivalently in this case, not to contain a non-abelian free subgroup. We give a simple condition on the matrix ensuring that it is abelian. This can be thought of as an effective Tits alternative on centralizers in $\sf{GL}(n,\mathbb{Z})$. We apply these criteria to the conjugacy problem in certain arithmetic groups preserving a non-degenerate $\mathbb{Q}$-bilinear form, such as integral symplectic groups. We derive an effective solution to the conjugacy problem in such groups when given matrices satisfy the above criterion. This solution is based on effective solutions to the conjugacy problem in $\sf{GL}(n,\mathbb{Z})$ by Eick-Hofmann-O'Brien and to an orbit problem for polycyclic groups, by Eick and Ostheimer.

The soliton-like coherent structure (SCS), which has been verified to exist in both transitional and turbulent boundary layers1-4, still poses a challenge in the understanding of its formation and behavior. In our previous study (Niu et al.5), the SCS was also found to exist in the transitional wake flow behind a sphere. In present study, the formation and evolution of the SCS is further investigated at four Reynolds numbers by numerical simulation. The results show that at the early stage of the turbulence transition, the SCS appears as a form of wave packet during the Tollmien-Schlichting (T-S) wave stage. With the increase of the Reynolds number, the SCS reaches its maximum amplitude downstream where the velocity discontinuity occurs. This position is located after the breakdown of the T-S wave and the three-dimensional structure is formed. Then, the SCS conserves its shape and amplitude over a long distance downstream. The relationships among the SCS, the spikes, the vortex structures, and the high-shear layers are further analyzed. It is found that the SCS in the wake flow has similarities to the phenomena observed in boundary layer flows during the turbulent transition. The vortex structures and high-shear layers mostly wrap around the border of the SCS. The vortex structure is considered to be as a consequence of the development of the SCS rather than its cause.

Given a porous set $E\in \mathbb{R}^d$ and a dyadic lattice $\mathcal{D}$, we refine the Carleson packing condition and the sparseness property for the dyadic cover $\mathcal{D}_E=\{Q \in \mathcal{D}: \: Q \cap E \neq \varnothing\}$. We study the inverse problem, when a Carleson family $\mathcal{S} \subset \mathcal{D}$ generates the porous set $E$ such that $\mathcal{S} \subset \mathcal{D}_E$.

Let $p$ be a prime number, and let $Δ_1,Δ_2 < 0$ be two coprime fundamental discriminants. When $p$ splits in $\mathbb{Q}(\sqrt{Δ_1})$ and $\mathbb{Q}(\sqrt{Δ_2})$ the height pairings of the corresponding CM divisors on $X_{\mathrm{spl}}^+(p)$ were determined by Gross--Kohnen--Zagier [GKZ87]. When $p$ is inert, we determine the arithmetic intersection numbers of the corresponding divisors on $X_{\mathrm{ns}}^+(p)$ at all finite primes. The key point of our analysis is at the prime of bad reduction $p$: to determine the intersection numbers at $p$, we provide a moduli interpretation for the smooth locus in the regular model of $X_{\mathrm{ns}}^+(p)$ over $\mathrm{Spec}(\mathbb{Z})$ constructed by Edixhoven--Parent [EP24].

We study Hamiltonian systems near a compact symplectic Morse-Bott minimum. Our first result shows that if the flow is Zoll (that is, it induces a free circle action) along a sequence of energy levels converging to the minimum, then the Hessian of the Hamiltonian in the symplectic normal directions must be compatible with the restriction of the symplectic structure to the normal bundle (that is, its representing endomorphism is a complex structure of the symplectic normal bundle). For our second result, we specialize to magnetic systems on closed manifolds with symplectic magnetic form. In this setting, if the system is Zoll along a sequence of energy levels converging to the minimum, then the metric is compatible with the magnetic form and therefore defines an almost Kähler structure. We show that a natural curvature quantity, consisting of the holomorphic sectional curvature corrected by a term measuring the non-integrability of the almost complex structure, must be constant. In particular, we obtain a dynamical characterization of complex space forms among Kähler manifolds. Together, these results establish strong rigidity of systems which are Zoll at energies close to a Morse-Bott minimum, in the symplectic and in the magnetic settings.

Ensuring ciphertext indistinguishability is fundamental to cryptographic security, but empirically validating this property in real implementations and hybrid settings presents practical challenges. The transition to post-quantum cryptography (PQC), with its hybrid constructions combining classical and quantum-resistant primitives, makes empirical validation approaches increasingly valuable. By modeling IND-CPA games as binary classification tasks and training on labeled ciphertext data with BCE loss, we study deep neural network (DNN) distinguishers for ciphertext indistinguishability. We apply this methodology to PQC KEMs. We specifically test the public-key encryption (PKE) schemes used to construct examples such as ML-KEM, BIKE, and HQC. Moreover, a novel extension of this DNN modeling for empirical distinguishability testing of hybrid KEMs is presented. We implement and test this on combinations of PQC KEMs with plain RSA, RSA-OAEP, and plaintext. Finally, methodological generality is illustrated by applying the DNN IND-CPA classification framework to cascade symmetric encryption, where we test combinations of AES-CTR, AES-CBC, AES-ECB, ChaCha20, and DES-ECB. In our experiments on PQC algorithms, KEM combiners, and cascade encryption, no algorithm or combination of algorithms demonstrates a significant advantage (two-sided binomial test, significance level $α= 0.01$), consistent with theoretical guarantees that hybrids including at least one IND-CPA-secure component preserve indistinguishability, and with the absence of exploitable patterns under the considered DNN adversary model. These illustrate the potential of using deep learning as an adaptive, practical, and versatile empirical estimator for indistinguishability in more general IND-CPA settings, allowing data-driven validation of implementations and compositions and complementing the analytical security analysis.

We study the continuous-time structure of the difference-of-convex algorithm (DCA) for smooth DC decompositions with a strongly convex component. In dual coordinates, classical DCA is exactly the full-step explicit Euler discretization of a nonlinear autonomous system. This viewpoint motivates a damped DCA scheme, which is also a Bregman-regularized DCA variant, and whose vanishing-step limit yields a Hessian-Riemannian gradient flow generated by the convex part of the decomposition. For the damped scheme we prove monotone descent, asymptotic criticality, Kurdyka-Lojasiewicz convergence under boundedness, and a global linear rate under a metric DC-PL inequality. For the limiting flow we establish an exact energy identity, asymptotic criticality of bounded trajectories, explicit global rates under metric relative error bounds, finite-length and single-point convergence under a Kurdyka-Lojasiewicz hypothesis, and local exponential convergence near nondegenerate local minima. The analysis also reveals a global-local tradeoff: the half-relaxed scheme gives the best provable global guarantee in our framework, while the full-step scheme is locally fastest near a nondegenerate minimum. Finally, we show that different DC decompositions of the same objective induce different continuous dynamics through the metric generated by the convex component, providing a geometric criterion for decomposition quality and linking DCA with Bregman geometry.

We introduce a variational formulation of the homogeneous Boltzmann equation, with hard-sphere cross section, which selects the unique energy conserving solution. We prove that this solution arises from the microscopic dynamics, namely Kac's walk, and we establish the propagation in time of entropic chaoticity, under the minimal assumption that the initial distribution is entropically chaotic.

We develop a novel multi-layer predictor-feedback to achieve exact compensation of state-dependent input delay of general nonlinear integro-differential equations. The system of interest is an unconventional mixed Partial Differential Equation (PDE)-Ordinary Differential Equation (ODE) system, in which a nonlinear ODE is actuated through an inhomogeneous advection PDE. Moreover, the propagation speed of the PDE depends on a moving window integral of the ODE state. The two above features are not addressed yet in standard PDE backstepping-based predictor-feedback designs. Unlike the conventional Lyapunov-based approaches used in the field, our stability and well-posedness analysis rely on the characteristic method and a fixed-point argument. Both of our designs achieve global asymptotic stability (GAS) in the supremum norm of the PDE and ODE states under the mild assumption that the nonlinearity in the PDE governing equation is uniformly Lipschitz continuous. The transport speed, governed by the integral of the ODE state, models systems such as production or queuing processes in which the state of a finite buffer-namely, the inventory level-determines the production or service rate. Numerical simulations demonstrate the effectiveness of the proposed control design for buffer-regulated production lines and queuing systems, ensuring asymptotic stability under a locally safe softened bang-bang feedback law that preserves the positivity of both the system state and the actuation signal.

We present and analyze a discontinuous Galerkin method for the numerical solution of a class of second-order linear mixed-type partial differential equations, i.e. equations that change their nature from elliptic to hyperbolic through the computational domain. Well-posedness of the discrete problem is established via coercivity in an energy norm, achieved through the Morawetz multiplier technique. We derive $hp$-a priori error estimates in the energy norm, which we use to prove convergence rates for standard and quasi-Trefftz polynomial spaces. Numerical experiments validate the theoretical results.

The Krasnosel'ski\uı-Mann algorithm is a well-known method for finding fixed points of a nonexpansive mapping with strong theoretical guarantees. However, there are practical large-scale problems to which this algorithm cannot be applied. Here, to resolve the issue caused by the computational difficulty of the mapping, we define a computable mini-batch stochastic mapping, which is a unbiased estimator of the nonexpansive mapping, and implement it in the Krasnosel'ski\uı-Mann algorithm. We show that the algorithm with increasing batch sizes converges almost surely to a fixed point of the nonexpansive mapping. We also perform a convergence rate analysis on the algorithm.

Most quantum divergences derive their structure from classical f-divergences or Renyi-type constructions, a dependence that obscures several quantum geometric effects. We introduce a quantum relative-alpha-entropy that extends Umegaki's relative entropy while falling outside the f-divergence class. The proposed divergence exhibits a nonlinear convexity property, which yields a generalized convexity result for the Petz-Renyi divergence for alpha greater than one, complementing the known convexity for alpha less than one. It is additive under tensor products, invariant under unitary transformations, and depends only on the relative geometry of quantum states rather than their absolute magnitudes. Using Nussbaum-Szkola-type distributions, we also establish an exact correspondence of this divergence with classical relative-alpha-entropy. This reveals relative-alpha-entropy as a fundamentally geometric notion of quantum distinguishability not captured by existing divergence frameworks.