We study dynamic resource allocation in a multicore computing system with a fixed number of processing cores and a stream of {\it malleable} jobs. Each job may adjust its level of parallelism during execution, allowing adaptive redistribution of resources across concurrently active jobs. Jobs belong to one of two observable classes, each characterized by a distinct speed-up function with unknown parameters. The objective is to learn a core-allocation policy that minimizes the long-run mean number of jobs in the system, equivalently the mean response time in steady state. \noindent To address this uncertainty, we develop an iterative learning-and-control framework. The system alternates between estimating the unknown speed-up parameters from observed job completions and solving the associated Markov decision process (MDP) to update the allocation policy. Within each job class, cores are shared equally among active jobs; the fraction of capacity assigned to each class is obtained from the MDP formulation of \cite{berg2017}, evaluated at the current parameter estimates. We construct a maximum likelihood estimator based on state-dependent inter-departure times and prove its strong consistency under a fixed allocation policy. We further propose two learning algorithms that combine this estimation step with dynamic programming-based policy updates, and illustrate their through numerical experiments.

Given a complex reductive group $G$, a representation $V$ of $G$ and a Borel-stable subspace $M \subset V$, we consider the associated Steinberg-type variety $Z$. We prove that, under a certain condition on $(V,M)$, called gluability, the equivariant Borel-Moore homology or $K$-theory of $Z$, equipped with the convolution product, is obtained as the intersection of two copies of the nil-Hecke algebra inside its localization. We also provide a description of these new algebras in terms of poles and residues. Similar results are obtained when $G$ is replaced by its loop group. This generalizes results of Ginzburg, Kapranov and Vasserot describing the affine Hecke algebra and DAHA, as well as a result of Teleman and Gannon--Webster that realizes certain Coulomb branches by gluing two copies of the universal centralizer.

Patterson-Sullivan measures encode the distribution of orbits of discrete group actions near the boundary. In this paper, we prove a global shadow lemma for Patterson-Sullivan measures associated to relatively Morse subgroups of higher-rank semisimple Lie groups. The estimate is uniform for shadows centered at arbitrary points in a Gromov model, including points deep in the cuspidal part. This extends the global shadow lemma of Stratmann-Velani for geometrically finite real hyperbolic groups. As applications, we obtain uniform local estimates for Patterson-Sullivan measures, and we give sufficient conditions under which these measures agree, up to scale, with the Hausdorff measure defined by the associated visual quasi-metric.

We propose a signature-based method to solve the optimal market-making problem under a mean-variance criterion. By exploiting signature linearization techniques, we reduce the market-making problem to a pseudo-linear optimization over the expected signature of an augmented market path, and we develop a signature algorithm named Sig-REINFORCE to learn the optimal bid and ask quotes. We test our method in two scenarios, in which market-order arrivals follow either a Poisson or a self-exciting Hawkes process, and we benchmark it against a Proximal Policy Optimization (PPO) baseline.

We introduce a second-order approximation to the compressible atmospheric Euler equations with gravity that is invariant domain preserving and well-balanced with respect to rest states. The approximation is built upon discrete auxiliary states derived from a hydrostatic reconstruction of the density. These auxiliary states, together with an affine shift of the numerical state, provide local bounds needed for maintaining well-balancing and invariant domain preserving properties of the method. The numerical method is then verified and validated with analytic solutions, well-balancing tests, and typical benchmark problems for atmospheric flows.

We prove an optimal dimension-free sparsification theorem for suprema of centered Gaussian processes. Given a bounded set $T\subseteq\mathbb{R}^n$, we show that the supremum of the canonical Gaussian process on $T$ can be $L^2$-approximated by the supremum of a shifted subprocess indexed by only $\exp(O(1/\varepsilon^2))$ points, with error at most $\varepsilon$ times the Gaussian width of $T$. In particular, the size of the approximating process is independent of both the ambient dimension and the cardinality of the original index set. This improves a recent sparsification theorem of De, Nadimpalli, O'Donnell, and Servedio (2026) by an exponential factor, and we show that the dependence on $\varepsilon$ is tight up to constants in the exponent. As consequences, we obtain an exponentially improved junta theorem for norms over Gaussian space and sharpen results on learning, property testing, and polyhedral approximation of convex sets under the Gaussian measure. The proof is based on an interpolation argument that combines Sudakov's minoration with the Brascamp--Lieb inequality.

The Helmholtzian of a graph $G$ is the Hodge $1$-Laplacian $L_1=L_1^{\mathrm{up}}+L_1^{\mathrm{down}}$ of its clique complex, built from the triangle--edge and edge--vertex boundary operators $\partial_2$ and $\partial_1$. Problem~5.5 of Lu, Shi, Stanić, Wang and Wang asks whether $λ_{\max}(L_1)=μ_1(G)$ for every graph $G$, where $μ_1(G)$ is the largest Laplacian eigenvalue; by the Hodge decomposition this is equivalent to $λ_{\max}(L_1^{\mathrm{up}})\leμ_1(G)$. We recast it as a question about the complement of $G$: localizing $L_1^{\mathrm{up}}$ on the cycle space of $K_n$ turns it into the inequality $λ_{\min}(\bar L|_{Z_1})\ge a(\overline{G})$, where $\bar L$ is the up Laplacian of the missing triangles of $G$ and $a(\overline{G})=n-μ_1(G)$ is the algebraic connectivity of the complement. From this viewpoint, we prove the unconditional bound \[ λ_{\max}\!\big(L_1^{\mathrm{up}}(G)\big)\ \le\ μ_1(G)+\frac13\big(n-μ_1(G)\big), \] which refines the integrality ceiling $λ_{\max}(L_1^{\mathrm{up}})\le n$ of Duval and Reiner and is sharp exactly when that ceiling is attained. We then isolate the single sharp inequality, on the dense part of $\overline{G}$, that stops the method short of Problem~5.5, and we show that the localization, the bound, and this obstruction all persist for the up Laplacian of an arbitrary finite simplicial complex, in every dimension.

The paper extends previous results on the nth root problem to a large class of Hilbert-space operators, namely, the class of all normaloid operators with normaloid parts, which includes the paranormal operators, and also the $k$-paranormal operators. It is shown that if a normaloid operator with normaloid parts has a normal nth power, then it is normal.

We develop a partial differential equation formulation of the Behrens-Fisher problem for two independent normal samples with unknown and unequal variances. An orthogonal decomposition separates mean and residual components (corresponding to the centered within-sample variation left after removal of the mean directions) and recasts the studentized difference of sample means as a scale-invariant geometric constraint. This reduction transforms the distributional problem into the evaluation of spherical wedge probabilities, which are identified with harmonic measure and with the value at the origin of a Laplace-Dirichlet boundary value problem. From this framework, we derive exact finite-sample representations for the cumulative distribution function and the probability density function in terms of beta functions, with dependence only on the sample sizes and the variance ratio. These representations place the Behrens-Fisher law in a standard special-function form that is directly accessible in widely available commercial software -- including Microsoft Excel -- thereby facilitating distributional evaluation and quantile computation. We also obtain a Gegenbauer separation-of-variables expansion for the associated harmonic extension and its threshold derivative, with coefficients in closed Beta-Gamma form, and derive sharp tail expansions with explicit leading constants and higher-order corrections.

For a family of graphs $\mathcal F$, a graph $G$ is said to be $\mathcal F$-free if it contains no member of $\mathcal F$ as a subgraph. A wheel graph $W_k$ is a graph on $k+1$ vertices formed by joining a new vertex to all vertices of a $k$-cycle. Given an integer $k\ge 3$, we consider the problem of determining the maximum number of triangles in a $W_{\geq k}$-free graph, where $W_{\geq k}=\{W_\ell: \ell \geq k\}$. The case $k=3$ was raised by Gallai, who proposed a conjecture for this case (see Erdős [5]. Gallai's conjecture was disproved by Zhou [17] and independently by Füredi, Goemans, and Kleitman [9]. In this paper, we study the case $k=4$. Namely, for every integer $n\ge 3$, we determine the maximum number of triangles in an $n$-vertex $W_{\geq 4}$-free graph and characterize all extremal graphs.

We propose and analyze a structure-preserving numerical method for the $2\tfrac{1}{2}$-dimensional (2.5D) electron magnetohydrodynamics system with fractional dissipation on the periodic torus. The method works directly with the magnetic field components and combines this component formulation with the gradient recovery operator of [T. Chu, H. Guo, and Z. Zhang, SIAM J. Numer. Anal., 63 (2025), pp. 23--53]. We establish discrete energy stability for a semi-implicit structure-preserving formulation and use an explicit-Hall integrating-factor implementation for efficient computation on periodic grids. The fractional dissipation is treated exactly in Fourier space, and the in-plane divergence constraint is enforced by a spectral Hodge projection. Numerical experiments demonstrate second-order spatial convergence and stable Hall-driven dynamics across several benchmark tests.

We study the Cauchy--Dirichlet problem for parabolic complex Hessian equations on Hermitian manifolds and on bounded strictly m-pseudoconvex domains. In the smooth setting, we prove global existence and uniqueness of classical solutions under the presence of an admissible parabolic subsolution, by establishing a priori estimates up to the parabolic boundary. The estimates combine parabolic boundary techniques for complex Hessian equations with interior second order estimates and a blow-up argument. We then develop a general pluripotential framework for degenerate right-hand sides with L^p densities, p>n/m, and bounded Cauchy--Dirichlet data. Since the usual automorphism and Walsh-type arguments do not directly apply in a variable Hermitian background, we use approximation by smooth data, balayage, parabolic Perron envelopes, and a continuous obstacle approximation based on Harvey--Lawson--Plis subequation theory. The resulting solution is continuous for positive time, locally uniformly Lipschitz and semi-concave in time, and continuous up to the initial slice when the initial datum is continuous. We also prove a parabolic comparison principle via time regularization, Riemann sum approximations, and mixed Hessian inequalities.

In this paper, we study the transition matrix between the PBW basis and the canonical basis for the negative part of the quantized enveloping algebra of the Kronecker quiver from a geometric viewpoint. Building on Lusztig's geometric construction of the canonical basis, we construct sheaf-complex realizations of PBW basis elements by means of flag sheaf complexes over the strata $X(α,m)$ of representation varieties. Our first goal is to give a geometric description of the simple constituents appearing in the restrictions of these flag sheaf complexes to the strata $X(α,m)$. This allows us to compare the PBW-type sheaf complexes with the simple perverse sheaves $IC(X(α),L_χ)$ arising in Lusztig's construction. Using this description together with a purity result for the relevant $\mathbb{F}_q$-structures, we obtain another proof that the elements defined by Lusztig's perverse sheaves indeed form a basis of the composition algebra.Our second goal is to make the transition coefficients between the PBW basis and the canonical basis geometrically explicit. More precisely, we show that these coefficients are governed by the multiplicities of local systems in the restrictions of intersection cohomology complexes to smaller strata. As a consequence, the transition matrix from the canonical basis to the PBW basis is upper triangular with diagonal entries equal to $1$, and its coefficients admit a direct geometric interpretation. In particular, in the Kronecker quiver case we recover the triangularity of the transition matrix and obtain positivity properties of the corresponding coefficient polynomials.

Working within Constructive Morse Set Theory (CMST), we introduce axioms for a new notion, jst Pp, intended to capture what it means for P to prove, or justify, p under the BHK interpretation of intuitionistic logic. Since it makes no distinction between terms and formulae -- every term is also a formula, and vice versa -- CMST is well suited to our axiomatic development of justification theory within set theory itself. After stating our axioms for jst Pp, we derive many consequences thereof. In particular, we show that (with certain restrictions) our axioms for jst Pp align with the intended BHK interpretations of the axioms of intuitionistic logic.

In this paper, we study probabilistic representations for stationary HJBI-type variational inequalities with bilateral constraints. We provide two complementary stochastic representations.The first representation is obtained through an augmented infinite-horizon two-player zero-sum stochastic differential game (SDG). By enlarging the control spaces with two additional stopping symbols, the obstacle terms are incorporated into the running payoff. Using the framework of infinite-horizon stochastic recursive differential games, we show that the resulting lower and upper value functions are the unique bounded viscosity solutions of the corresponding HJBI variational inequalities. The second representation is given by a two-player zero-sum mixed control--stopping SDG. In this formulation, each player chooses both a continuous control and a stopping decision, and the payoff is defined by a BSDE with a random terminal time. To make the stopping component compatible with the Elliott--Kalton strategy framework, we introduce nonanticipative stopping strategies depending on the opponent's control process. The proof is based on penalized infinite-horizon SDGs coupled with their own value functions, together with dynamic programming arguments and stability estimates for backward semigroups. We prove that the value functions of the mixed control--stopping game coincide with the unique bounded viscosity solutions of the bilateral HJBI variational inequalities.

Tight equalities between symmetric information and entropy production in driven steady states remain elusive. We show that they are forbidden by a parity selection rule for rotation-driven linear nonequilibrium steady states. Whenever the relaxation and diffusion matrices commute, the snapshot mutual information between two time slices is exactly even under drive reversal, and parity violation rises linearly in the commutator norm when alignment is broken. Full isotropy strengthens this to drive-independence, and the planar mutual information takes the closed-form value of about 0.145 nats. Under the same alignment, the entropy production is exactly quadratic in the drive, and its prefactor admits an explicit closed form in the traces and determinant of the two matrices. The orthogonality of even and odd sectors leaves only one-sided thermodynamic-uncertainty bounds. The rule rests on the rotational symmetry of the drift alone and survives heavy-tailed isotropic stable noise with tail index below two, where variance-based bounds become vacuous. A falsifiable test is proposed on an electrical Brownian gyrator augmented for independent drive control with circuit-level stable-noise injection.

Let $\cpsi_{2,a}(n)$ denote the number of $(2,a)$-colored Frobenius partitions of weight $n$, where the two rows have prescribed length difference. We study the two cases $a=0$ and $a=1$ and connect them with minimal-excludant statistics on bipartitions. Let $σ\mex_2(n)$ be the sum of the Lin--Liu bipartition minimal excludants over all bipartitions of $n$, and let $E_2(n)$ be the number of bipartitions whose two component minimal excludants are equal. For all $n\geq 0$, we give a combinatorial proof of \[ \cpsi_{2,0}(n)=2σ\mex_2(n) \qquad\text{and}\qquad \cpsi_{2,1}(n)=2σ\mex_2(n)-E_2(n). \] These identities give direct combinatorial interpretations of two-colored Frobenius partition functions in terms of bipartition minimal-excludant sums.

Twisted quantum loop algebras are a generalization of twisted quantum affine algebras in Drinfeld new presentation. The Hall algebras of Geigle--Lenzing's weighted projective lines are used to realize (untwisted) quantum loop algebras of simply-laced type associated to star-shaped graphs by Schiffmann and Dou--Jiang--Xiao. In this paper, we use the semi-derived Ringel-Hall algebras of more general weighted projective lines to realize the twisted quantum loop algebras associated to the valued star-shaped graphs, including the twisted quantum affine algebras in Drinfeld new presentation.

Variance-based finite-time Lyapunov exponents (vFTLEs) provide a stochastic analogue of deterministic FTLE by measuring the covariance of stochastic arrival locations. Existing PDF-based formulations compute this covariance by solving a Fokker--Planck equation for each initial point, which becomes expensive when the diagnostic is required on a dense grid. In this work, we develop a moment-based Eulerian approximation to vFTLE in the small-noise regime. Starting from a stochastic trajectory expansion about the deterministic flow, we derive a closed covariance equation for the leading stochastic displacement. By embedding this trajectory-wise covariance dynamics into physical space, we obtain an Eulerian transport--reaction equation for a symmetric covariance tensor field. The covariance associated with each initial point is recovered by evaluating this tensor field at the deterministic arrival location, and a moment-based vFTLE is then defined from its largest eigenvalue. The proposed method replaces a family of Fokker--Planck solves by the evolution of a single covariance tensor field, requiring only $d(d+1)/2$ scalar fields in $d$ dimensions. It also retains directional information through the eigenvectors of the covariance tensor, allowing the dominant directions of stochastic spreading to be visualized. We establish the leading-order consistency of the method with PDF-based vFTLE in the small-noise limit, clarify its relation to scalar stochastic sensitivity, and show how the same covariance equation connects process-noise spreading with deterministic deformation. In particular, deterministic FTLE is recovered, up to an additive constant, from an isotropic initial covariance when no process noise is present, while continuous process noise produces a time-integrated deformation covariance.

We show, for example, that if one samples \[\frac{\log p}{2\log(1+(p-1)^{-1})} \cdot n^2(1 + o_{n\to \infty}(1))\] points in $\mathbb{F}_p^n$ at random then asymptotically almost surely this set intersects every quadratic hypersurface. We furthermore show that this is tight in that sampling $o_{n\to\infty}(n^2)$ fewer points almost surely fails to intersect some quadratic hypersurface. Our main result is a sharp threshold for the following problem: how many points in $\mathbb{F}_p^n$ does one need to randomly sample to almost surely intersect every algebraic set defined by at most $s$ polynomials each of degree at most $k$? As an application we improve lower bounds in the random Szemerédi theorem in $\mathbb{F}_p^n$, in particular obtaining a leading constant which grows as the threshold for what is considered a `dense' set in Szemerédi's theorem shrinks.

Establishing stability guarantees for dynamical systems on Lie groups is a fundamental challenge, as classical Lyapunov methods developed for Euclidean spaces do not directly transfer to curved geometries. In this paper, we propose a framework for learning maximal Lyapunov functions for systems evolving on the special orthogonal group $\mathsf{SO}(n)$. Theoretically, we introduce a neural Lyapunov architecture based on the logarithmic map with proven approximation capabilities, and we formulate the learning problem via a Zubov-type characterization of the maximal region of attraction. A key technical contribution is the derivation of explicit, numerically tractable formulas for the derivative of the logarithmic map, enabling training through a two-phase algorithm that balances computational efficiency and accuracy. Empirically, we validate the approach on a low-dimensional nonlinear system.

Aided by the AI tool GPT5.5 Pro, we provide a counterexample to a conjecture made by Ramachandra and Natarajan (2025) [Pairwise independent correlation gap, Operations Research Letters, 107255, 6040].

We develop a capacitary approach to semilinear elliptic inequalities on weighted graphs with a potential. More precisely, we study the nonexistence of nontrivial nonnegative solutions of \[ Δu+w(x)u+v(x)u^σ\le0 \qquad\text{in }V, \] where \((V,ω,μ)\) is a connected, locally finite weighted graph, \(Δ\) is the associated graph Laplacian, \(σ>1\), \(v>0\), and \(w\) is a real-valued potential. The potential term is handled by means of a positive solution \(H\) of \(ΔH+wH=0\), which transforms the operator \(Δ+w\) into the \(H\)-Laplacian associated with a new weighted graph. Our main nonexistence criterion is formulated directly in terms of cut-off functions and the regions where their \(H\)-Laplacian is controlled. Unlike metric criteria based on pseudo-metric annuli, our formulation determines the capacitary sets from the support of the \(H\)-Laplacian estimates for the cut-off functions. We provide an example showing that our result applies in situations not covered by previous nonexistence criteria based on structural lower bounds or pseudo-metric annular volume estimates. We also show that the growth exponent in our capacitary condition is sharp by constructing an example for which the condition fails by an arbitrary power \(R^\varepsilon\), while a positive nontrivial solution exists.

The derivative nonlinear Schrödinger equation is a fundamental model for the propagation of nonlinear dispersive waves in, for example, plasma physics and nonlinear optics. In this work, we consider this model on the one-dimensional torus and study a filtered explicit Fourier integrator for the corresponding periodic problem. After applying a periodic gauge transformation, we consider a frequency-truncated model and its filtered exponential-Euler discretization. The main difficulty comes from the derivative cubic nonlinearity in the periodic setting, since local smoothing is unavailable and resonant interactions are stronger than in the non-periodic case. To address this issue, we develop a discrete Bourgain-space framework adapted to the gauge-transformed equation. For initial data $u_0 \in H^s(\mathbb{T})$ with $1/2 < s \le 5/2$, we prove that the numerical error is of order $\mathcal{O}(τ^{s/2-1/4})$ in $H^{1/2}(\mathbb{T})$, where $τ$ denotes the employed time step size. Numerical experiments confirm the predicted convergence behavior and demonstrate the effectiveness of the filtered scheme for rough solutions.

Exact renormalisation techniques are important and powerful, particularly for inflation-generated systems. We review recent results in this direction. We recall the necessary notions for inflation systems and show the renormalisation principle, which allows us to obtain exact values of highly erratic functions, such as window covariograms. We apply these techniques to compute the diffraction pattern of the new monotile tilings with arbitrary precision. We also recall a recent invariant for system with pure-point spectrum, the orbit separation dimension, and its relation to renormalisation. Lastly, we recall results beyond the pure-point spectrum setting and show how renormalisation and Lyapunov exponents can be used to exclude the presence of absolutely continuous part of the spectra.

We study optimal control of a system with multiple decision makers who share a common hidden state and receive fully decentralized observations through identical channels. The dynamics of the hidden state and the cost incurred by the agents depend on the agents' actions only through their empirical distribution. In the limit problem with infinitely many agents, the problem reduces to a single agent control problem where the agent affects the hidden state dynamics via the conditional law of the actions given the past values of the hidden state process. We formulate this problem as a deterministic measure valued control problem over the space of policies and provide a dynamic programming recursion. We first show that for the limiting problem randomization over the control actions is necessary for optimality. However, randomization over the selection of policies (i.e., mixture policies) is not required. We then show that the optimal symmetric policies designed for the infinite population problem are near optimal for the finite population problem. In particular, we establish convergence rates that decay with number of agents as $\frac{1}{\sqrt{N}}$, and grow exponentially with the memory length used in the policy.

For the Zakharov system in four space dimensions, we prove that all solutions inside the potential well of the ground states are global and scattering in the energy space, with no other restriction such as symmetry. The proof has already been reduced by [3] to ruling out the existence of a minimal non-scattering solution that is precompact along some trajectory. This paper carries out the final step in the proof, namely we exclude the possibility of precompact solutions inside the potential well by combining two distinct arguments depending on the motion of trajectory.

We extend the optimal transport theory for signed measures supported on Ahlfors-regular fractal sets (Bwo'Nyahre et al., 2026) to allow a controlled dimensional distortion between source and target. A penalization term $\varepsilon Φ(d_s(x) - d_t(y))$ -- where $Φ$ is a fixed smooth strictly convex function and $d_s, d_t$ are the local Hausdorff dimensions of the fractal supports -- is added to the transport cost on inter-sign regions, with~$\varepsilon \ge 0$ controlling the tolerance for distortion. Under hypotheses H1--H7, we prove: the existence and uniqueness of an optimal transport map~$T^{\varepsilon}$ for every~$\varepsilon > 0$; coupled Monge--Ampère equations with a distortion correction term, generalizing the classical Brenier--Caffarelli equation; a double Legendre--Fenchel characterization of the optimal potentials, giving a complete variational description of the transport in each of the four sign regimes. The double Legendre--Fenchel system (Theorem~4.2) is the central contribution: it shows that the optimal potentials are the unique fixed points of a system of conjugacy equations, one per transport regime, and it provides the foundation for numerical algorithms and asymptotic analysis.

We study the space of one-point functions on the torus for a possibly nonrational $C_2$-cofinite vertex operator algebra $V$ by relating it to a trace object of the subcategory of projective objects in the representation category of $V$. We identify the dual of the trace space with symmetric functions on the endomorphism algebra $E$ of a projective generator. Motivated by the Gainutdinov-Runkel conjecture, recently established using different methods by Gui and Zhang, we present a complementary representation-theoretic approach based on Arike-Nagatomo pseudo-traces. In this framework, we prove surjectivity of the Gainutdinov-Runkel map from symmetric functions on $E$ to one-point functions. Under the additional assumption of separated conformal weights modulo $\mathbb{Z}$, we also prove injectivity, using projective-cover techniques inspired by Huang.

We study the Monge--Kantorovich optimal transport problem between two signed measures~$μ$ and~$ν$ on convex compact subsets of~$\mathbb{R}^d$, with a positional penalization function~$λ(x, y)$ that modulates the cost of inter-sign transport. Using four independent positive measures~$(π^{++}, π^{+-}, π^{-+}, π^{--})$ as decision variables, we prove that the admissible set~$\mathcal{A}(μ, ν)$ is weakly-$*$ compact and non-empty if and only if $μ^+(X) = ν^+(Y)$ and~$μ^-(X) = ν^-(Y)$. Strong duality is established via the Kantorovich minimax theorem, yielding a new compatibility condition on~$λ$ at the intersection of inter-sign supports. The penalization~$λ$ is shown to be Lipschitz and to admit Alexandrov second derivatives almost everywhere. Modified Monge--Ampère equations governing inter-sign transport maps are derived in the Alexandrov sense, with well-posedness characterized by $σ\det(D^2_{yx}Λ) e > 0$. The classical Brenier equation is recovered in the limit~$λ\to 0$.