Papers

We prove that the Dirichlet $p$-torsion function $u_p$ converges to the distance to the boundary $d$ at the rate $O(1/p)$ in the uniform norm, on every bounded domain and with a constant depending only on the dimension and the diameter. The upper bound comes from a radial barrier with its pole at a boundary point, which is an admissible comparison function for $p>n$ and needs no boundary regularity; the lower bound comes from an inscribed ball. On the ball the error equals $(1+\log n)/p+O(p^{-2})$, so the order $1/p$ cannot be improved in general. Under a uniform exterior ball condition an annular barrier gives $u_p\le K_\Omega^{1/(p-1)}d$ for every $p>1$, with $K_\Omega$ explicit in the dimension, the diameter and the exterior radius; on convex sets the constant is the diameter, and a multiple of $d$ is a supersolution whenever $-\Delta d$ has a positive distributional lower bound. On $C^2$ domains, the same barrier and the classical gradient maximum principle give $\norm{\nabla u_p}_{L^\infty}^{p-1}\le K_\Omega$. When $-\Delta d$ is a measure of finite total variation, which we prove for $C^1$ domains with a uniform exterior ball, the gradients converge in $L^q$ at the rate $O(p^{-1/2})$ for $q\le2$ and $O(p^{-1/q})$ for $q\ge2$; these exponents are not claimed to be sharp. The explicit ball profile shows that the gradients do not converge uniformly.

A complete one-ended three-manifold with strictly positive Ricci curvature is constructed such that $$ \limsup_{R\to\infty}\frac1R\int_{B(p,R)} Scal\,dV=+\infty. $$ The construction combines an explicit toric lens carrying a large scalar-curvature integral with a three-dimensional angular pair of pants and an adaptive sequence of hybrid blocks.

The Donoho–Stark uncertainty principle asserts $N^d\le|E|\,|\Sigma|$, and the additive energy uncertainty principle of Aldahleh–Iosevich–Iosevich–Jaimangal–Mayeli–Pack refines it to $1\le|E|\cdot\|1_\Sigma\|_{U^2}^{4/3}$. It has been conjectured that the natural higher-order analogue \[ 1\le |E|\cdot\|1_\Sigma\|_{U^k}^{2^k/(k+1)} \] holds for every $k\ge 2$, the coset case giving equality. We show that this fails for every $k\ge 6$. The counterexample is the two-point signal $f=\delta_0-\delta_1$ on $\mathbb{Z}_3$, for which the left-hand quantity equals $2/3(2(k+1))^{1/(k+1)}$; this is $<1$ precisely when $k\ge 6$, and tends to $2/3$.

We construct tie-symmetrised extensions $\widetilde D$ and $\widetilde R$ of the unnormalised Hoeffding and Blum–Kiefer–Rosenblatt functionals $D$ and $R$ and prove, for every real-valued bivariate law, the exact nonnegative decomposition $ \tau^{*}(X,Y)=12\widetilde D(X,Y)+24\widetilde R(X,Y), \qquad \widetilde D,\widetilde R\geq0, \qquad \widetilde R(X,Y)=0\quad\Longleftrightarrow\quad X\perp\!\!\!\perp Y. $ For atomless margins the components reduce to $D$ and $R$, whereas in the presence of ties the corresponding classical identity $\tau^{*}=12D+24R$ can fail. The components arise from the Hoeffding projections of a symmetrised pair kernel. For the four strict/non-strict boundary versions $D_{\epsilon,\eta}$ and $R_{\epsilon,\eta}$, we establish the sharp comparisons $ 9\widetilde D\geq\sum_{\epsilon,\eta}D_{\epsilon,\eta}, \qquad 9\widetilde R\geq\sum_{\epsilon,\eta}R_{\epsilon,\eta}. $ Consequently, $\tau^{*}\geq0$ for every bivariate law, and the Bergsma–Dassios conjecture is settled: $\tau^{*}=0$ if and only if $X\perp\!\!\!\perp Y$. The boundary comparisons also yield the universal bound $\tau^{*}\geq(8/3)R$. Finite-table arguments, including an exact sum-of-squares certificate, establish the boundary inequalities, and weak-order quantisation transfers them to arbitrary laws. The factor $9$ in each comparison and the coefficient $24$ in $\tau^{*}\geq24\widetilde R$ are sharp. The usual permutation test is consistent against every fixed dependent alternative, without assumptions on the margins.

We prove a uniform shifted anticoncentration theorem for the hafnian of a real Gaussian Gram matrix under an explicit condition on the row dimension. After normalization by its root mean square, the law has a bounded continuous density, maximal at zero, and every interval has probability bounded by an explicit coefficient times its radius. Under suitable growth conditions on the row dimension, this coefficient grows at most polynomially in the hafnian order, meaning half the dimension of the Gram matrix. We also compute the exact second moment. The proof exploits the perfect matching structure, combining conditional Gaussian representations, row suspension, and bilinear interpolation to control an inverse moment of the conditional variance. At fixed hafnian order, a rescaled limit as the row dimension grows yields corresponding bounds for the hafnian of a real symmetric Gaussian matrix with independent entries above the diagonal.

We characterize locally finite sets \(T\subset[0,\infty)\) that preserve exact observability from nonnegative integer times for every bounded positive operator \(A\) and bounded observation operator \(C\). The Stein identity reduces universality to uniform upper and lower bounds on radial Stein–Pick cones over all finite spectral models. Writing \(N_T(R)=\#(T\cap[0,R])\), we prove that universal Bessel preservation is equivalent to \(N_T(R)=O(R)\). The conditions \(0\in T\) and \(N_T(R)\asymp R\) characterize universal observability preservation under \([A,C^*C]=0\), as well as uniform bound preservation for each fixed number of spectral values. However, we construct a clustered set satisfying these conditions whose optimal sampled lower bounds tend to zero as the spectral count grows, despite fixed integer observability bounds. The same set destroys the frame property of a singly generated positive diagonal Carleson frame while preserving its Bessel property. When natural density exists, universality is equivalent to \(0\in T\) and \(0<d(T)<\infty\). More generally, containing zero and a subset of positive finite natural density suffices under linear upper counting. For sets containing zero, universality is invariant under sublinear differences of counting functions. For general normal operators, universal Bessel preservation is instead equivalent to uniformly bounded counts on unit intervals; under a fixed logarithmic sector condition, it is again characterized by linear upper counting.

A criterion for the convexity of a two-dimensional domain is proved in terms of the solution of a boundary value problem for the modified Helmholtz equation.

Let $X$ be a $(q-1)$-connected rational space of finite type, where $q\ge2$, with homotopical nilpotency $nil_h(X)=n\ge1$. If $H^{>N}(X;Q)=0$ and $N\le q(n+3)-3$, we prove that $Hnil(X)=n$. More precisely, the sufficient bound is $N\le a_n(M_X)+2q-3$, where $a_n(M_X)$ is the first nonzero internal cohomology degree of $(M_X^+)^{n+1}$ in the minimal model. A family of examples proves that the constant $-3$ in this refined bound is optimal; optimality of the uniform bound is not asserted. The proof uses two stages of adjoining primitives to an ideal of the fixed minimal model.

In [Finite groups with a small number of cyclic subgroups. Czechoslovak Mathematical Journal 75.3 (2025): 839-851], the authors classified all the groups having 13 cyclic subgroups. In this paper, we give a new proof for the classification of all finite groups having exactly 13 cyclic subgroups.

A time fractional extension of the Hunter Saxton equation is examined, in which the temporal derivative of u_x is replaced by a Caputo derivative of order 0 < alpha <= 1. This modification introduces memory effects into a model traditionally associated with director field dynamics in nematic liquid crystals. By employing a fractional separation of variables strategy together with the exponential spatial profile phi(x) = exp(b+x), which cancels the nonlinear structure exactly, the governing nonlinear partial differential equation is reduced to a fractional relaxation ODE, whose closed form solution is the one-parameter Mittag Leffler function. The exact analytic solution is derived algebraically from the separation process. This appears to be the first exact closed form solution of the Caputo time fractional Hunter Saxton equation. The result is validated symbolically using the MathHandbook computer-algebra system. Quantitative analysis in both the fractional (0 < alpha < 1) and classical (alpha -> 1) limits demonstrates how fractional-order memory slows temporal relaxation relative to the exponential baseline. The solution provides a reliable benchmark for numerical schemes and clarifies how fractional dynamics influence nonlinear wave propagation in orientationally ordered fluids.

Physical and synthetic models may describe complementary aspects of the same PDE-governed system while receiving different, possibly fragmented, observations. We propose Bi-Objective HYCO (Bi-HYCO), a cooperative framework that retains both representations and their local observational objectives while coupling their predicted states at unlabeled interaction points. These points contain no measurements and do not augment the data; they provide a communication mechanism in the common state space. The two criteria form a vector-valued objective, and weighted scalarizations provide computational realizations. For the deterministic shared-observation algorithm with fixed interaction points, we prove sufficient decrease and finite length of the whole alternating sequence, which converges to a mixed critical point under the stated Kurdyka-Lojasiewicz-type assumptions. Elliptic transmission and two-dimensional Navier-Stokes experiments assess parameter and state reconstruction, noise and scalarization effects, and PINN/XPINN references. Ablations show that removing state interaction while retaining aggregation deteriorates parameter recovery in the tested configurations, particularly for Navier-Stokes.

Various persistent invariants have been developed in recent years. In this paper, we develop a method based on homological algebra for deriving central limit theorems for persistent Betti numbers associated with $\mathbb{R}$-indexed chain complexes constructed from a homogeneous Poisson point process of unit intensity on $\mathbb{R}^d$. Our method also applies to Gibbs point processes. This algebraic method reduces the verification of the required conditions to checking properties of the kernels and cokernels of add one maps between chain complexes. As an application, we recover the known central limit theorem for persistent Betti numbers arising from simplicial complex filtrations. In addition, we prove a central limit theorem for persistent Betti numbers of $\ell_p$-Vietoris-Rips simplicial homology, including blurred magnitude homology of random geometric graphs. We also establish central limit theorems for relative $\ell_p$-Vietoris-Rips homology and, in particular, for magnitude Betti numbers of random geometric graphs. These results suggest that our algebraic method can be applied more broadly to derive central limit theorems for persistent Betti numbers.

This paper presents novel extremum seeking algorithms inspired by RMSProp dynamics and based on Lie bracket approximation techniques. To inherit the behavior of the classical continuous-time RMSProp algorithm, we introduce an extremum seeking scheme that exploits second-order Lie brackets, allowing the excitation of squared gradient terms in the resulting Lie bracket system. Moreover, we propose two modified RMSProp-type algorithms that can be implemented using only first-order Lie bracket approximations, thereby reducing the complexity of the extremum seeking control design. The three resulting extremum seeking algorithms are compared through numerical simulations.

Let $(M^n,g_0)$ be a closed smooth Riemannian manifold, with $n\geqslant 3$. We prove short–time existence and uniqueness for the "critical" Ricci–Bourguignon flow \begin{equation} \partial_t g=-2\operatorname{Ric}_g+\frac{R_g}{n-1}g\,. \end{equation} Up to a constant rescaling of time, this is the Schouten flow, since the right–hand side of the equation is $-2(n-2)$ times the Schouten tensor. At the critical parameter, however, the principal symbol of the linearization of the operator that we obtain after the usual "DeTurck modification" still has a zero eigenvalue. We resolve this degeneracy by adjoining an independent scalar unknown $r$, intended to represent the scalar curvature, and considering an "extended" system consisting of a strictly parabolic equation for the metric coupled to a transport–reaction equation for $r$ with a curvature–square forcing. The extended system belongs to the class of nonlinear composite parabolic–hyperbolic systems studied by Vol'pert and Khudyaev~\cite{VolpertKhudyaev1972}. Then, a standard compact–manifold adaptation of their theorem yields a unique smooth solution of the extended system. Finally, the difference $R_g-r$ satisfies a homogeneous linear parabolic equation, so the scalar–curvature constraint "propagates" and the original critical Ricci–Bourguignon flow is recovered. This settles the short–time existence and uniqueness problem at the critical “Schouten” value $\rho=1/(2(n-1))$, which was left open by the previous Ricci–Bourguignon theory in~\cite{CatinoEtAl2017}.

We prove a new estimate on Jacquet-Whittaker function for $\mathrm{GL}_n(\mathbb R)$, assuming that the Langlands parameters are purely imaginary and well-spaced. This gives an upper bound for the global sup-norm of the Whittaker function that matches the lower bound of Brumley-Templier in the exponent up to a linear error in $n$.

For classical knots, many characteristics are difficult to determine from arbitrary diagrams of the knot. Bridge number is of this type. It is possible to have a bridge presentation of a knot in which the number of bridges is only locally minimal, as shown by Ozawa and Takao. Here, we give a method for beginning with an arbitrary classical knot diagram having \(n\) crossings and producing a locally minimal bridge presentation of the knot with size \(O(n^2)\). This method requires only polynomial time and space. For some classes of knots, any local minimum is also the global minimum, which was shown for the trivial knot by Otal. Because the unknot is the only knot with bridge number \(1\), unknot recognition is in \(P\).

We introduce a nonlinear Poissonian fluctuating Boltzmann equation whose noise encodes both the fluctuations and the path large-deviation rate function of the underlying hard-sphere gas. Unlike the Gaussian-noise equations commonly studied in fluctuating hydrodynamics, the present equation raises a new difficulty: a Poisson random measure produces jumps that may destroy the nonnegativity of the solution. To address this problem, we construct a finite-dimensional coarse-grained jump process and prove its well-posedness and nonnegativity. For each fixed mesh, this process satisfies a good path large-deviation principle. We then fix a regularized velocity cutoff and study the asymptotic behavior of the discrete rate functions as the mesh is refined. On a class of biased regular paths, their limit is the corresponding cutoff Boltzmann large-deviation rate function associated with the hard-sphere gas. This establishes the consistency of the coarse-grained fluctuating Boltzmann model with the underlying particle system at the level of path large deviations.

We prove that, for every $k\ge2$ and every sufficiently large odd $n$, the induced size–Ramsey number of $C_n$ satisfies \[ \widehat R_{\mathrm{ind}}(C_n;k)=e^{\Theta(k)}n. \] This improves the previous upper bound of $e^{O(k\log k)}n$ and resolves a conjecture of Bradač, Draganić and Sudakov. We obtain this result by proving a stronger form of their odd-hole gadget conjecture: we construct a graph with $e^{O(k)}$ edges whose every $k$-edge-colouring contains a monochromatic induced odd cycle of length $O(\log k)$. Our proof develops a spread-based approach to finding induced cycles. The key idea is to place spread probability weights directly on families of candidate cycles and control their possible chords through a quadratic chord score. Combining this with random sparsification and hypergraph containers yields a sparse host graph that works for every $k$-edge-colouring.

This paper studies the spectral properties and nonlinear asymptotic stability of a family of finite-time self-similar blow-up solutions to the three-dimensional Keller–Segel system constructed by matching interior and exterior profiles within the framework of matched asymptotic expansions. For every sufficiently large matching index $n$, the full linearized operator around the stationary state $U_n$ in self-similar variables is analyzed on $L^2(\mathbb R^3)$. Sturm zero counting in the radial mode, a wave operator that reduces the nonlocal $l=1$ equation to a local equation, and a Mellin–Newton quadratic form for all $l\ge2$ show that, after the scaling and translation modes and the finitely many genuinely unstable radial modes are removed, the remaining spectrum is separated from the imaginary axis by a positive distance. In addition, the optimal lower bound on the real parts of the spectrum is $1/4$ in every mode $l\ge2$. On the stable subspace, an exponentially decaying semigroup and a modified energy equivalent to the $L^2$ norm are constructed, and the logarithmic asymptotic decay rate of the semigroup norm is proved to equal the stable spectral gap. Finally, modulation equations, $H^2$ energy estimates, control of the scaling derivative, and Brouwer's no-retraction theorem yield nonradial nonlinear asymptotic stability of the corresponding self-similar blow-up solutions after the initial coefficients in the finitely many unstable radial directions have been chosen suitably.

Iima and Yoshino asked for an ideal $I$ in $S=k[x_1,x_2,\ldots]$, with $\operatorname{deg} x_i=i$, and a monomial order such that $S/I\cong k[x_i:i\equiv\pm1\pmod5], \operatorname{in}(I)=(x_i^2,x_ix_{i+1}:i\geq1).$ We construct such an ideal and monomial order over every field $k$ of characteristic different from $5$ containing an element $c$ with $c^2+c=1$. The ideal has an explicit infinite homogeneous reduced Gröbner basis. A five-periodic syzygy derived from a pentagon identity proves that all basis relations belong to $I$ and supplies standard representations for the non-coprime critical pairs. Triangular elimination establishes the graded quotient isomorphism. Together, the quotient and initial ideal descriptions yield the partition form of the first Rogers-Ramanujan identity. In each weighted degree, a perfect matching in the support of the normal-form matrix gives a bijection between the two partition classes. We formalize the complex specialization in Lean 4 using Mathlib and our set-based theory of infinite Gröbner bases, including the reduced basis, the graded quotient isomorphism, and the partition-matching theorem.

We consider a system of $N$ interacting bosons in a rotating harmonic trap in $\mathbb{R}^2$, where the two-body interaction is attractive and scaled as $N^{2\alpha}U(N^\alpha x)$ with $0<\alpha<1/12$, and the three-body interaction is repulsive and scaled as $N^{4\beta}W(N^\beta x,N^\beta y)$ with $0<\beta<1/24$. In the mean-field limit, the ground state energy is effectively described by a rotating cubic-quintic nonlinear Schrödinger functional. We analyze the collapse regime where the two-body coupling $a$ approaches the critical value $a_*$ and the three-body coupling $b$ tends to zero. The NLS ground states blow up with a universal profile given by the optimizer of the Gagliardo-Nirenberg inequality, and the energy satisfies $E^{\mathrm{NLS}} = (1-\zeta/4+o(1))\mathcal{Q}{\rm{pot}}\ell_n^2$ with $\zeta\geqslant0$ determined by the relative rates of $a_n\to a_*$ and $b_n\searrow0$. From the many-body theory, we rigorously justify this effective description: the quantum ground state energy converges to the NLS energy with the same asymptotic expansion in the collapse regime, and the many-body ground states exhibit complete Bose-Einstein condensation onto the universal blow-up profile.

We prove that a nonconstant polynomial $f\in\mathbb{C}[x_1,x_2,x_3]$ is cylindrical (that is, $f\in\mathbb{C}[x_1,x_2]$ after an invertible linear change of coordinates) if and only if its bordered Hessian determinant vanishes identically. The same conclusion holds over $\mathbb{R}$. We also give explicit counterexamples to this criterion in every dimension $n\ge4$.

We study slice regular functions on the real division algebras $\mathbb{H}$ and $\mathbb{O}$ from the point of view of equivalence under automorphism and conjugation actions. Motivated by recent orbit-theoretic descriptions in terms of the invariants $(Tr,N,cdiv)$, and by the quaternionic theory of $*$-conjugation by semiregular functions, we investigate whether a slice regular function can be replaced by a canonical representative in its equivalence class. The normalization considered in this paper is algebraic: we look for representatives whose isolated nonreal zeros are aligned in a single complex slice. We call such a representative a model when it belongs to the strong-equivalence class of the original function. We prove that purely vectorial functions always admit models, while in general strong equivalence is too rigid and such representatives need not exist. We therefore introduce semi-models, which preserve the symmetrization and the real and spherical part of the zero set, but are allowed to leave the original strong-equivalence class. Our main result proves that every slice regular function $f$ with $N(f)\not\equiv 0$ admits a semi-model. The construction clarifies the different roles of the invariants $(Tr,N,cdiv)$ and the distinction between strong equivalence, weak equivalence, and semiregular conjugacy.

Let \(\beta(G)\) denote the minimum size of a feedback edge set of a digraph \(G\), and let \(\gamma(G)\) denote the number of unordered pairs of nonadjacent vertices. Motivated by the Chudnovsky–Seymour–Sullivan conjecture for \(3\)-free digraphs, we study the corresponding feedback-edge problem for bipartite digraphs. In the bipartite setting, \(\gamma(G)\) is taken to count only nonadjacent pairs with ends in distinct partite sets. We prove that every \(4\)-free bipartite digraph \(G\) satisfies \(\beta(G)\le \gamma(G)/2\). We also determine the exact Turán number of \(2k\)-free strong bipartite digraphs with partite sets \(X\) and \(Y\): if \(|X|,|Y|\ge k+1\), then the maximum number of edges is $$(|X|-(k-1))(|Y|-(k-1))+2k-2.$$ Finally, for the extremal case \(k=2\), we analyze the structure of \(4\)-free strong bipartite Turán digraphs and prove the sharper bound \(\beta(G)\le \gamma(G)/3\) for all such digraphs. This constant is attained by a natural balanced three-block construction.

Let $R_n$ be the infimum of the inradii of $\{z:|p(z)|<1\}$ over monic degree-$n$ polynomials whose zeros lie in the closed unit disk. We prove $nR_n\to\pi/2$, matching the asymptotic obstruction supplied by $z^n-1$. We first establish the exact universal radius $2^{1/n}-1$ for disks centered at zeros, which recovers the $(\log2)/n$ bound. Small inradius then forces radial concentration of the zeros and decay of their low reciprocal moments. These estimates give an entire limit with a modulus reflection identity; a second rescaling produces an exponential tangent and strict sublevel disks of every radius below $\pi/2$. The proof is accompanied by a Lean~4 formalization and a step-by-step source index.

The stochastic Landau–Lifshitz–Gilbert (sLLG) equation is a strongly nonlinear stochastic PDE with a non-convex pointwise constraint arising in the theory of micromagnetics. We analyse a fully discrete, structure-preserving finite element approximation of the sLLG equation with coloured multiplicative Stratonovich noise on a bounded interval. The method utilises continuous piecewise affine finite elements, mass lumping, and midpoint time discretisation to preserve the unit-length constraint exactly at the finite element nodes. Under suitable regularity assumptions on the initial data and the noise, we establish uniform higher-moment stability and develop an error analysis for the scheme. The analysis exploits the geometric structure of the equation and the stochastic midpoint discretisation. For every $\gamma\in(0,\frac12)$, we prove first-order spatial convergence and temporal convergence of order $\gamma$ in the natural discrete energy norm, locally in mean square on events of arbitrarily large probability and, consequently, in probability. To the best of our knowledge, this is the first convergence-rate result for a fully discrete structure-preserving finite element scheme solving the stochastic Landau–Lifshitz–Gilbert equation.

In this paper, we study when the restriction of an irreducible representation of a reductive Lie group $G$ to a reductive subgroup $H$ is discretely decomposable. T.\ Kobayashi gave a necessary condition for discrete decomposability in terms of associated varieties. In a previous paper, we proved that the condition is also sufficient if $(G, H)$ is a symmetric pair. The purpose of this paper is to prove the sufficiency for any reductive subgroup $H$ under some hypothesis on $G$ (e.g., $G$ is linear). As an application, we show that discrete decomposability is equivalent to $\mathfrak{h}$-admissibility when the compact factor of $\mathfrak{h}$ is maximal in some sense.

Let $R$ be a commutative noetherian local ring with residue field $k$. Denote by $\operatorname{\mathsf{D^b}}(R)$ the bounded derived category of finitely generated $R$-modules. In this paper, we introduce the notion of ranks of triangulated categories and study Verdier quotients of $\operatorname{\mathsf{D^b}}(R)$ from the viewpoint of this invariant. Our main result determines the rank of the Verdier quotient $\operatorname{\mathsf{D^b}}(R)/\operatorname{\mathsf{thick}}(R\oplus k)$ in terms of the ranks of the categories of maximal Cohen-Macaulay modules on the punctured spectrum.

We show that there is a Hadamard space in which the closed convex hull of a set containing merely three points is noncompact, by providing an AI-generated construction. The seemingly innocuous problem of finding a compact or finite set with this property, or showing that none exists, was recorded by Gromov and periodically revisited by researchers in metric geometry, functional analysis, fixed-point theory, and geometric group theory. Nevertheless, it remained unanswered for over three decades. The construction itself is based on building an inductive sequence of CAT(1) metric graphs and taking the Euclidean cone over the limiting space. We discuss the history of this problem, the main ideas and details of the construction, and its relationship with existing techniques.

In this paper, we present a high-order, stable summation-by-parts (SBP) finite difference method for solving the Vlasov-Maxwell system in a 2D2V phase space. Central SBP operators for the advection terms are not stable when the solution becomes non-smooth and fine-scale filamentary structures develop, as is typical in high-dimensional Vlasov-Maxwell simulations. To address this issue, the method is stabilized using high-order upwind SBP operators. High-order explicit Runge-Kutta methods are employed for time integration. We prove that the fully discrete scheme exactly conserves mass and preserves momentum up to truncation error. Furthermore, we present a matrix-free implementation of the method on modern GPU architectures. A range of challenging benchmark problems is solved to demonstrate the accuracy, robustness, and performance of the proposed scheme.