Papers

Assume $G$ is a graph and $k$ is a positive integer. Let $f:V(G)\to \mathbb{N}$ be defined as $f(v)=\min\{k,d_G(v)\}$. If $G$ is $f$-choosable, then we say $G$ is degree-truncated $k$-choosable. The degree-truncated choice number of $G$ is $\operatorname{ch}^{\text{\st{d}}}(G) = \min\{k: G \text{ is degree-truncated $k$-choosable}\}$. For a family $\mathcal{G}$ of graphs, $\operatorname{ch}^{\text{\st{d}}}(\mathcal{G}) = \max\{\operatorname{ch}^{\text{\st{d}}}(G):G \in \mathcal{G}\}$. Let $\mathcal{P}$ denote the family of 3-connected non-complete planar graphs. Richter asked in 2008 whether $ch^{\text{\st{d}}}(\mathcal{P}) \le 6$. In 2025, Zhou, Zhu and Zhu answered this question in negative and proved that $8 \le ch^{\text{\st{d}}}(\mathcal{P}) \le 16$. This result was improved by Jiang, Xu, Xu, and Zhu, who proved that $9 \le ch^{\text{\st{d}}}(\mathcal{P}) \le 12$. In this paper, we further improve the result and prove that $10 \le \operatorname{ch}^{\text{\st{d}}}(\mathcal{P}) \le 11$. We conjecture that $\operatorname{ch}^{\text{\st{d}}}(\mathcal{P}) =10$, and we confirm this conjecture for those planar graphs $G \in \mathcal{P}$ for which the subgraph induced by vertices of degree at least 11 is 4-choosable.

We study non-trivial intersection problems for multi-partite hypergraphs, excluding the usual extremal examples determined by fixed vertices or fixed coordinates. Our first result determines the exact value of the non-trivial $t$-intersection problem in the symmetric product $[n]^r$ for $1\le t\le r-2$ and all $n\ge2$. Frankl and Nie proved a two-candidate formula for sufficiently large $n$ and conjectured it for all $n\ge 2$; our formula shows that the conjectured expression must be enlarged, in small ranges of $n$, by additional ball-type terms arising from the Frankl families. Our second result concerns intersecting families in general products $X_1\times\cdots\times X_r$, where $|X_i|=n_i$, with no common vertex. Let $m_0(1,n_1,\ldots,n_r)$ denote the largest size of such a family. We show that this number is equal to the maximum of $\sum_{X\in \mathcal{D}}\prod_{i\in X}(n_i-1)$ over all downsets $\mathcal{D}\subseteq 2^{[r]}$ such that $\bigcup_{X\in \mathcal{D}}X=[r]$ and no two members of $\mathcal{D}$ have union $[r]$. This finite reduction separates the intersection obstruction from the part sizes and yields explicit fully asymmetric formulas for $r=4,5,6$.

We study some local and global properties of solutions of $-\Delta u- m\abs{\nabla u}^q-e^{u}=0$ in a punctured domain $\Omega\setminus\{0\}$, or in an exterior domain of $R^N$, $N\geq 2$, where $m$ is a positive parameter and $q>1$. We study particularly the local behaviour of solutions with an isolated singularity or the asymptotic behaviour for solutions defined in an exterior domain, and also the existence of solutions with the behaviours previously described. These behaviours change drastically according $q$ is smaller or larger than $2$. Many results are obtained by introducing various dynamical systems associated to the equation.

For a finite graph $H$ and a positive integer $h$, the $h$-blowup $H^{(h)}$ of $H$ is the graph obtained by replacing each vertex of $H$ by a set of size $h$ and each edge by a complete bipartite graph between the corresponding sets. We prove that, for every $H$, there exists a constant $h_*(H)$ such that whenever $h\ge h_*(H)$ and $n$ is sufficiently large, every $n$-vertex graph maximizing the number of induced copies of $H^{(h)}$ is a blowup of $H$. This refines the asymptotic result of Hatami, Hirst and Norine and settles the question posed by Bollobás, Egawa, Harris and Jin in 1995.

We study a stochastic optimal control problem for McKean-Vlasov stochastic differential equations (SDEs) with common noise, where the dynamics depend on the conditional law of the state. We derive a stochastic maximum principle of Peng type without imposing convexity assumptions on the control domain. In comparison to the standard McKean-Vlasov case, the maximum principle for the common noise case contains a third adjoint state, which is needed to dualize all second-order Lions derivatives in the Taylor expansion of the cost functional. The additional adjoint state is given by a conditional McKean-Vlasov backward SDE. All three adjoint states together allow for a complete linearization of all contributions in the second-order expansion, including interactions between conditionally independent copies of the first variational process. As part of our analysis, we also prove a general well-posedness result for conditional McKean-Vlasov backward SDEs.

The goal of this short note is to prove a unified abstract regularity lemma which recovers Szemerédi's graph regularity lemma, Green's arithmetic regularity lemma, and a regularity lemma for Boolean functions as direct corollaries.

We survey a line of works studying semimartingale reflected Brownian motion in a quadrant, covering both the non-degenerate and degenerate settings. Two main situations are emphasized: the recurrent case, where an invariant measure exists, and the transient case, where the central objects are Green's functions (potential measures). These measures typically arise from Kolmogorov forward equations. For transient or killed models one is also interested in the Martin boundary of the process and, consequently, in all positive harmonic functions, which satisfy Kolmogorov backward equations. Depending on the geometry and parameters of the model, these harmonic functions often admit probabilistic interpretations in terms of absorption, escape, or drift to infinity. All these measures and functions are studied through kernel functional equations satisfied by their Laplace transforms. Several ways of solving these equations are reviewed, each leading to different types of results. Following the analytic approach developed for quarter-plane random walks by Fayolle, Iasnogorodski and Malyshev, a key preliminary step is the analytic continuation of the relevant Laplace transforms onto the complex algebraic curve defined by the zero set of the kernel. Carleman boundary value problem techniques then yield explicit contour-integral representations for the Laplace transforms. In special parameter regimes, Tutte's invariant method provides integral-free formulas and a sharp classification of the transforms according to their algebraic-differential complexity. Singularity analysis combined with saddle-point methods carried out on the kernel algebraic curve, produces precise two-dimensional asymptotics. Finally, in the degenerate setting, the compensation approach provides an alternative constructive method, allowing one to build the densities as an infinite series through explicit iterative corrections.

We characterize the boundedness and compactness of slice regular composition operators between quaternionic Fock spaces for the full range \(0<p,q<\infty\), without assuming that the composition symbol preserves a fixed complex slice. As applications of the same method, we also obtain corresponding criteria for weighted composition operators and for products of Volterra-type integral operators with slice regular composition operators. The main tool is a fixed-slice matrix realization of the regular product, which represents slice regular composition on a fixed complex slice through a holomorphic \(2\times 2\) matrix functional calculus. This representation reveals a genuinely quaternionic rigidity phenomenon: boundedness imposes affine restrictions on the eigenvalue functions of the associated matrix symbol rather than on the original symbol itself. In particular, the original symbol need not be affine, and affine eigenvalue functions alone do not characterize boundedness.

In this article we develop a quadratic energy which approximates the Canham-Helfrich energy for a tube-like surface with clamped boundary and area constraint. The energy is suited to the study of small deformations of biological membranes where the deformations are induced by point forces or point constraints due to the cytoskeleton or a phase dependent spontaneous curvature. Since the deformations we consider are small, we may assume that the surface of interest is a graph over a fixed, undeformed surface. A Lagrangian and the associated Euler-Lagrange equations for the graph are derived. Well-posedness of the Euler-Lagrange equations in suitable spaces is shown. Finally, we provide some illustrative numerical examples.

Physics-Informed Neural Networks inherently suffer from task interference because they rely on a shared parameter space to satisfy both governing differential equations and boundary conditions. We analyze this structural conflict using the Fisher Information Matrix to quantify the effective degrees of freedom ($d_{eff}$) in a physics-constrained model. Unlike the classical $d_{eff}$ which measures how many parameter directions are informed by data against a statistical prior, our $d_{eff}$ measures the dimension of the parameter directions unconstrained by the differential operator. For operators with finite-dimensional kernel, we show that $d_{eff}$ converges to the kernel dimension exactly, independent of network width, depth, or activation function, recasting it from a fit diagnostic into a structural invariant of the underlying continuous operator. For operators with infinite-dimensional kernel, $d_{eff}$ instead measures the network's finite-dimensional representational bandwidth for that kernel rather than recovering an integer invariant. Importantly, $d_{eff}$ also serves as an a priori structural diagnostic. Driving $d_{eff}$ of a well-posed problem to zero certifies that the physics and boundary constraints have absorbed the network's free directions. Building on this characterization, we introduce subspace projection strategies for boundary adaptation. Rather than retraining from scratch, we project parameter updates into the null space of the pre-trained physics operator so that new boundary conditions are satisfied without disturbing the learned physics. Gradient-based fine-tuning can match or exceed this but needs more wall-clock time and tuning, whereas subspace projection delivers near-equivalent quality in seconds to minutes. We validate on linear and nonlinear operators, demonstrating accurate adaptation to initial and boundary shifts and unencountered constraint types.

By now, several approaches have been proposed to endow phase-field fracture models with the ability to describe crack nucleation under multiaxial stress states. These include techniques for splitting the free energy, direct modifications of the phase-field driving or resisting forces that sacrifice the variational structure of the problem, and the introduction of additional internal variables, such as plastic strains or other nonlinear strains. In this paper, we propose a fundamentally different strategy for incorporating arbitrary elastic domains into phase-field fracture models, formulated within the variational framework of generalized standard materials. The proposed approach relies on letting the dissipation potential depend on the current state of the material. In this way, the variational structure of the problem is preserved, while elastic degradation and the strength criterion remain two distinct and independently controllable aspects of the material response. Simple yet representative models are presented and thoroughly discussed to demonstrate the effectiveness of the proposed methodology. The resulting evolution of the elastic domain is investigated in both strain and stress spaces. Moreover, numerical simulations demonstrate a range of crack nucleation processes under multiaxial loading conditions for various analytical strength surfaces. This work paves the way for future developments and applications in several directions.

Energy decay rates for solutions of the damped wave equation on the torus are known to be influenced by the geometry of the damped set and the growth properties of the damping. In this paper we produce lower bounds on energy decay rates for a class of damping which are positive on a superellipse and grow polynomially like the distance to the boundary of the superellipse. The energy decay rates we obtain depend explicitly on the exponent used to define the superellipse and the polynomial power. We show these rates are sometimes optimal. The proof adapts quasimodes from $y$-invariant damping using a simplification of the usual normal form argument.

We study quantum algorithms for nonlinear and stochastic homogenization via a Young-measure based linear programming (LP) formulation, which lifts the nonlinear problem to a linear one in higher dimensions by treating the microscale, the gradient, and possible random variables as independent variables, thereby capturing effective macroscopic quantities without directly resolving fine-scale oscillations. The resulting LP is large but structured, and its high-dimensional nature creates regimes in which quantum LP solvers outperform direct classical solvers: in the deterministic setting, polynomial quantum speedup arises when moderate homogenized accuracy suffices; in the stochastic setting, encoding all random realizations simultaneously in a single LP yields a quantum square-root reduction in stochastic sampling cost that grows with the number of random variables. Regularity or sparsity of the Young measure may further extend these advantages to fine-scale accuracy. Numerical experiments on one- and two-dimensional benchmarks confirm the correctness of the Young-measure LP formulation.

This paper develops equivariant basic cohomology for Lie groupoids equipped with weak actions of Lie groups. The weak action is encoded by a Kan fibration over the classifying groupoid, and the basic complex of the fiber is shown to carry the structure needed for Weil and Cartan models. The construction is compared with Bott–Shulman–Stasheff cohomology, where the equivariant theory is obtained from the quotient groupoid. For orbifolds, basic forms are interpreted as orbifold differential forms, and the resulting equivariant basic cohomology is used to formulate differential-geometric constructions such as equivariant integration and localization. The paper also studies the induced weak action on the inertia groupoid and uses it to define an equivariant refinement of the Chen–Ruan cohomology ring. In this framework the sectorwise equivariant cohomology, obstruction bundle, equivariant Euler class, Gysin maps and three-point functions are assembled into an equivariant Chen–Ruan product whenever the corresponding pairing is nondegenerate.

In many industrial and engineering applications, process performance is characterized by the ratio of two normally distributed quality characteristics. Monitoring such ratios is particularly challenging in short production runs, where conventional control charts often suffer from limited sensitivity due to the small number of available inspections. This paper proposes an exponentially weighted moving average (EWMA) control chart for monitoring the ratio of two normally distributed random variables under short production run (SPR) conditions. The statistical distribution of the ratio is first reviewed, adopting the corrected closed-form density of Nadarajah (2020) rather than the approximation used in earlier studies. The control limit of the proposed chart is calibrated to a prescribed in-control truncated average run length (TARL$ _0 $) over a finite horizon $ I $ of inspections, using a Markov-chain representation of the EWMA recursion. The detection performance of the chart is then assessed through a large factorial study covering the smoothing constant $ \lambda $, the in-control correlation $ \rho_0 $, the coefficients of variation $ (\gamma_X, \gamma_Y) $, the sample size $ n $, and the magnitude of the shift $ \tau $. Numerical results show that the proposed EWMA-RZ chart provides substantially better detection of small and moderate shifts than the recently developed Shewhart-type short-run ratio chart (ShRZ) of Tran et al. (2021), especially for $ |\tau - 1| \le 0.05 $. An illustrative example based on a beverage filling process is included to demonstrate the practical implementation of the method.

Let $G$ be a graph. The spectral radius of $G$ is the largest eigenvalue of its adjacency matrix. A matching of $G$ is a set of disjoint edges of $G$. The matching number of $G$ is the size of a maximum matching (i.e., a matching with maximum edges). The graph $G$ is called maximum matching covered if each edge of $G$ is contained in a maximum matching. In this paper, we give a sharp spectral radius condition for graphs with bounded matching number to be maximum matching covered.

We investigate circuit models, namely, modified nodal analysis (MNA) in the port-Hamiltonian framework. Based on this, the JR-decomposition for the numerical treatment would offer an energy conform splitting. However, for circuit models, the application of the standard JR-decomposition is restricted. To enable a JR-decomposition for MNA, we need to relax the decomposition. To this end, we introduce the enhanced JR-decomposition, which is particularly tailored to the application to circuits. We conclude with a numerical example that illustrates the applicability of the proposed approach as well as its convergence and structure-preserving properties.

In this work, we propose a novel intrinsic approach to the approximation of ergodic SDEs on Riemannian manifolds, which include Riemannian Langevin dynamics. In opposition to the standard extrinsic approaches such as penalization methods and projection methods, our methodology does not use embeddings or coordinates and only relies on natural geometric operations: geodesics, parallel transport,... We give a criterion for high order of accuracy for the invariant measure, develop new intrinsic numerical methods designed solely for sampling the invariant measure, and derive high order conditions using a new algebraic operation on exotic Lie-Butcher series. In the spirit of the Leimkuhler-Matthews method, our approach prioritizes long time sampling efficiency over finite time accuracy, and outperforms the previous extrinsic and intrinsic approaches in terms of cost for a given accuracy, which we illustrate with several numerical experiments.

In numerous industrial production settings, keeping track of the ratio formed by two normally distributed random variables is a task of considerable practical interest. The present work examines how measurement errors influence the behaviour of a pair of one-sided Synthetic control charts designed to monitor such a ratio (referred to here as Synthetic-RZ charts), with the analysis covering both the zero-state and the steady-state average run length ($ARL$). To incorporate measurement error into the operation of these charts, we adopt a linear covariate error model. We describe, step by step, how the parameters of the underlying model evolve as the process moves from an in-control to an out-of-control state, and we deliberately avoid the restrictive premise that the observed shift magnitude is unrelated to the measurement errors. The run length characteristics of the charts are obtained by means of a Markov chain formulation. A series of numerical experiments makes clear that measurement error erodes the detection capability of the charts. A particularly useful outcome of the investigation is that collecting several measurements on each inspected unit does not constitute an efficient remedy for the adverse influence of measurement error on the performance of the Synthetic-RZ charts.

In this paper, we determine the exact minimal number of curves in a filling $k$-system on an oriented surface of genus $g$ for any positive integers $k$ and $g$.

Monitoring the ratio of two correlated normal variables is increasingly important in statistical process control, since many quality characteristics are expressed in relative rather than absolute form. Memory-type ratio charts have mostly been developed for long production runs, while their finite-horizon counterparts rely on a fixed reference value $ k $ derived from a specified shift. Such fixed-$ k $ designs are not optimal at a given out-of-control magnitude and, in low-variability regimes, yield boundary solutions for which the in-control truncated average run length (TARL$ _0 $) is unattainable. This paper proposes an upper one-sided cumulative sum (CUSUM) control chart for the ratio $ Z = X/Y $ in short production runs, denoted CUSUM-RZ$ ^+ $ (RZ standing for the ratio $ Z $), with fully adaptive joint optimization of $ k $ and the decision interval $ h $. Given a target TARL$ _0 = I $ and a target shift $ \tau $, a bilevel problem calibrates $ h(k) $ by inner root-finding to satisfy the TARL$ _0 $ constraint and selects $ k^* $ by outer line search to minimize the out-of-control TARL$ _1 $. Both use a finite-state Markov-chain framework with an accurate ratio approximation; the inner step recovers boundary cases that fixed-$ k $ designs cannot. The chart is assessed through matched-horizon benchmarks against Shewhart-RZ, exponentially weighted moving average (EWMA-RZ), and fixed-$ k $ CUSUM-RZ$ ^+ $ charts, Monte Carlo robustness studies, and a Phase I estimation analysis. All memory-type charts outperform the Shewhart-RZ baseline; the adaptive design matches them under stable correlation and improves appreciably when correlation rises from Phase I to Phase II. It is insensitive to symmetric heavy tails yet mildly anti-conservative under contamination, and $ m \geq 100 $ subgroups keep the TARL$ _0 $ relative bias near 1%.

We study complete non compact gradient generalized m quasi Einstein manifolds with constant scalar curvature R <= 0, soliton function lambda > 0, and m > 1, where the coefficient mu = 1/m is constant. We introduce the weighted function v and prove it is subharmonic. This leads to four rigidity results, showing that the hypotheses are mutually inconsistent; consequently, no complete non compact gradient generalized m quasi Einstein manifold exists under the stated assumptions. We first show by a concrete example that if mu is allowed to be nonconstant, the rigidity conclusions fail even when all other hypotheses are satisfied. Therefore the constant mu condition is essential.

We establish an explicit upper bound on the number of stable Andrews-Curtis moves that convert thickenable balanced presentations of the trivial group to the standard one-generator presentation. We also present a proof that thickenable balanced presentations of the trivial group satisfy the (unstable) Andrews-Curtis conjecture.

Ensemble Kalman Inversion (EKI) is a derivative-free, ensemble-based method for inverse and optimization problems. A central limitation of its continuous-time dynamics is premature covariance collapse, which can restrict the directions accessible to the inversion and increase sensitivity to the initial ensemble. We introduce a second-order interacting particle system that combines the Kalman force with inertia, damping, attraction towards the ensemble mean, and short-range repulsion. We establish a generalized subspace property showing that nonzero initial velocities may enlarge the affine space accessible to the dynamics, while a finite-ensemble restriction remains. For linear inverse problems, we derive the mean and fluctuation dynamics, identify a regime in which fully collapsed configurations are linearly unstable, characterize asymptotic optimality on the subspace retained by the limiting covariance, and prove exponential decay for the associated frozen-covariance mean dynamics. Numerical experiments on high-dimensional linear and nonlinear inverse problems investigate subspace enlargement, discrepancy-based regularization, and ensemble collapse.

In this note, we generalize a 1950 result of P. Erd\H os on upper bounds of $k$-th moment of Romanoff type representation functions. As an application, we give a conditional proof of a recent conjecture of Y.-G. Chen on Romanoff type sumsets under the assumption of the Hardy-Littlewood conjecture.

We introduce pVR, a topological machine learning framework for alignment-free genomic sequence classification that combines $p$-adic numbers with topological data analysis. Each DNA sequence is encoded along two complementary axes: a $p$-adic distance on $k$-mer prefixes, which captures hierarchical positional structure, and a compositional $L_1$ distance on $k$-mer frequencies, which captures local sequence content. The two distances jointly parameterise a bi-filtered Vietoris–Rips complex, and per-sequence topological summaries from this bi-filtration serve as features for standard machine learning classifiers. We establish theoretical guarantees for the construction: stability under metric perturbations and invariance to the choice of prime, alongside a result that explains why a single $p$-adic axis is topologically uninformative and why the bi-filtration recovers nontrivial homology. On twelve genomic benchmarks ($28$ to $500$ sequences, $3$ to $7$ classes), pVR outperforms four established alignment-free baselines on three of six low-sample datasets, with gains of up to $21$ percentage points; it underperforms only on a SARS-CoV-2 variant benchmark whose point-mutation divergence violates the hierarchical assumption, and all methods saturate in the large-sample regime. pVR also outperforms zero-shot frozen embeddings from the 500M-parameter Nucleotide Transformer v2 by $6.7$ to $11.4$ percentage points on three low-sample benchmarks. The pVR codebase is publicly available at https://github.com/MAHI-Group/pVR.

We introduce a microlocal phase-space-filtered physics-informed neural network (PINN–TDPSF or Microlocal PINNFilter) framework for wave propagation on unbounded domains. The method combines a slabwise neural residual approximation of the interior evolution with a time-dependent phase-space filter applied in a buffer surrounding the physical computational domain. The central idea is to replace local artificial-boundary penalties by a phase-space radiation mechanism: a component is removed only when it is localized near the artificial boundary and its group velocity points outward. The proposed method is not intended to replace FFT, spectral, or split-step solvers for known-coefficient forward problems where such methods are available and highly accurate. Instead, it embeds the time-dependent phase-space filter into a residual-based neural framework. This coupling is useful when open-domain wave propagation must be combined with nonlinear residuals, sparse or off-grid observations, unknown coefficients, variable interior media, or other non-FFT-diagonalizable physics. Numerical experiments for linear Schrödinger propagation, potential scattering, anisotropic Schrödinger dynamics, nonlinear Schrödinger wave packets, soliton stress tests, linearized Euler waves, and sparse-data recovery of a localized acoustic defect show that the method reduces artificial reflection and wraparound, uses group velocity correctly in anisotropic media, preserves physically incoming branch components, and provides diagnostics when the assumptions behind outgoing-packet filtering are violated.

Quantum key distribution (QKD) is increasingly considered for deployment in realistic communication networks, where long distances, heterogeneous fiber infrastructure, and coexistence with classical traffic present substantial challenges. Here, we demonstrate trusted-node QKD between Linköping University and the Stockholm hub of the Swedish national quantum communication infrastructure over 270 km of deployed single-mode fiber, extended by a 33 km multi-core fiber (MCF) segment emulating a metropolitan access link, for a total distance of 303 km. The two sub-links use commercial QKD systems whose receivers are interfaced with external superconducting nanowire single-photon detectors, enabling operation at losses beyond those supported by standard internal gated-mode detectors. We operate the link while actively switching the QKD channel between two MCF cores, with co-propagating Ethernet traffic and injected broadband optical noise in the other cores. The results demonstrate the integration of commercial QKD into demanding, dynamically reconfigurable fiber infrastructure relevant to future hybrid quantum-classical networks. Finally, using the generated secret keys, we illustrate how limited and time-varying QKD throughput affects one-time-pad-protected image transmission: image fidelity depends strongly on the available QKD-generated key budget and the choice of compression algorithm, highlighting application-level challenges for QKD-based encryption in realistic scenarios.

This paper studies two questions on valued fields: the metric dimension induced by an absolute value, and the uniform openness of multiplication. For nontrivial non-archimedean absolute values, we prove that the metric dimension equals the density character. In the archimedean case, the metric dimension is 2 for subfields of $\mathbb{R}$, while for non-real subfields of $\mathbb{C}$ it is either 2 or 3, according to whether the field contains a non-real element together with its complex conjugate. We also show that multiplication is uniformly open on every valued field. Finally, we prove that this property is genuinely metric, not purely topological, even on $\mathbb{R}$ with a suitable compatible choice of metric.

We study natural automorphisms of the universal virtual braid group $UV_n(k)$. These automorphisms induce commuting involutions in the outer automorphism group and generate a subgroup isomorphic to $\mathbb{Z}_2^k\times\mathbb{Z}_2$. We then show that the two one-forbidden quotients of $UV_n(k)$ are isomorphic. Furthermore, we introduce the universal unrestricted virtual braid group $UUV_n(k)$ obtained by imposing simultaneously the two forbidden relations, and derive several structural properties inherited from the universal setting. Since the multi-virtual braid group $M_kVB_n$ is a quotient of $UV_n(k)$, the corresponding results for $M_kVB_n$ follow as consequences. In particular, for $k=1$ we prove that the quotients of $VB_n$ by the two forbidden relations are isomorphic and obtain structural properties for the unrestricted virtual braid group.