Papers

Extremely large-scale multiple-input multiple-output (XL-MIMO) is a promising technology for supporting automated guided vehicle (AGV) fleets in smart port terminals. However, the metallic container environment induces spatial non-stationarity, whereby each AGV is visible to only a subset of the array, referred to as its visibility region (VR). Unlike existing XL-MIMO models that assume user-independent VRs, we show that each AGV simultaneously acts as a communication user, a metallic scatterer, and a blocker, resulting in coupled user channels and VRs. We formulate this triple-role effect through a VR-coupled channel model and develop a VR-aware downlink beamforming framework based on masked weighted minimum mean-square error (WMMSE), where the masking operation exactly enforces VR support constraints while significantly reducing computational complexity. Simulation results in a realistic smart port scenario demonstrate more than a threefold sum-rate improvement over VR-unaware baselines, with the gains becoming increasingly pronounced as fleet density increases.

Let $X$ be a proper reduced scheme over a finite field $k$, let $\ell$ be a prime different from $\operatorname{char} k$, and write $\overline{X}=X\times_{k}\overline{k}$ for its base change to an algebraic closure $\overline{k}$ of $k$. Call a class in $\mathrm{H}^{2}_{\mathrm{\acute{e}t}}(\overline{X},\mathbb{Z}_{\ell}(1))$ Zariski-locally trivial if it vanishes on a Zariski-open cover of $\overline{X}$. We prove that the first Chern class map identifies $\operatorname{NS}(\overline{X})\otimes\mathbb{Z}_{\ell}$ with the group of Zariski-locally trivial classes whose image in $\mathrm{H}^{2}_{\mathrm{\acute{e}t}}(\overline{X},\mathbb{Q}_{\ell}(1))$ has weight zero. This is the finite-field analogue of a theorem of Barbieri-Viale–Rosenschon–Srinivas for proper seminormal complex varieties. In the finite-field setting neither seminormality nor irreducibility is needed.

We give a negative solution to Problem~13.23 of the Kourovka Notebook. We construct a torsion-free group $G$ of Hirsch length $14$ admitting a finite series \[ 1=G_0\triangleleft G_1\triangleleft\cdots\triangleleft G_{14}=G \] in which every $G_i$ is normal in $G$ and every factor is infinite cyclic, but such that $\Out(G)=1$.

This paper is concerned with the existence and the limiting behavior of measure attractors of distribution laws of the solution segment process for the McKean-Vlasov stochastic $p$-Laplace lattice system with time delay driven by Lévy noise. The nonlinear drift and diffusion terms are allowed to have superlinear growth. Due to time delay, the Skorohod metric space is employed to describe the trajectories of the solutions with jumps. We first prove the existence and uniqueness of càdlàg solutions for the lattice system, and then define a non-autonomous cocycle acting on the Borel probability measures in the Skorohod space. This cocycle is continuous in bounded subsets of the space of probability measures only when time is sufficiently large. We then prove the existence of pullback absorbing sets and the asymptotic compactness of the cocycle as well as the existence and uniqueness of pullback measure attractors. We finally investigate the limiting behavior of measure attractors of the lattice system as the noise intensity approaches zero, and establish the optimal convergence rate of singleton measure attractors in the Wasserstein distance of order $\theta$.

We prove the vanishing of the adjoint Bloch–Kato Selmer group of the Galois representations associated to regular algebraic automorphic representations of general linear groups over CM fields. A key novelty of our work is that we impose conditions only on the $p$-adic Galois representations, and not on their associated residual representations modulo $p$.

Colloquially speaking, `equivariant categories' refer to families of additive categories $\mathcal{A}(G)$ depending 2-functorially on a finite group $G$. We construct approximations of equivariant categories by Mackey 2-functors, both on the left and on the right. The idea is to enlarge $\mathcal{A}$ in a minimal way to make induction appear. These `mackeyfications' are inspired by Boltje's work with ordinary Mackey 1-functors. We also relate our left and right mackeyfications via a mark transformation. Finally we discuss examples.

We compute the untwisted part of the $\mathbb{C}^\times$-equivariant orbifold quantum cohomology of certain finite cyclic quotients of the minimal resolutions of Kleinian singularities, which we refer to as their foldings. We then formulate a conjecture for the full $\mathbb{C}^\times$-equivariant orbifold quantum cohomology, motivated by the Crepant Resolution Conjecture, and provide two pieces of supporting evidence. We also identify the resulting Frobenius structure with known Frobenius structures associated with the corresponding non-simply-laced root systems.

Let $F$ be a CM number field; then, to any cuspidal, regular algebraic automorphic representation of $\mathrm{GL}_n(\mathbf{A}_F)$ is associated a compatible system of $p$-adic Galois representations of the absolute Galois group of $F$. We prove that these representations are potentially semi-stable, in the sense of $p$-adic Hodge theory, and satisfy compatibility with the local Langlands correspondence, up to semi-simplification.

Quantum measurements can remain incompatible yet become impossible to certify when all probe states first pass through a quantum channel. We formulate this loss of certifiability as operational concealment and identify its exact finite-dimensional threshold. Under tomographically complete input access, a channel conceals some incompatible binary POVM pair if and only if its Heisenberg adjoint has nontrivial kernel. Thus, in this channel restricted single-use setting, rank loss rather than contraction is the universal transition for exact concealment: injective dissipative dynamics may render the effective measurements compatible while leaving exact concealment impossible. We quantify this distinction through a minimum-gain lower bound and show that regular time-local dynamics cannot produce exact concealment at finite time. We then classify all qubit channels whose adjoint fibres admit a positive unital retraction, reducing concealment to ordinary joint measurability of canonical representatives. For orthogonal projected qubit measurements we obtain an exact robustness formula and a strict separation from ordinary incompatibility robustness. Finally, quotient-space duality expresses this robustness as an experimentally accessible additive advantage in positive-payoff channel-output games. We also prove invariance under adjoining a passive finite-dimensional reference system for lifted target measurements, even when arbitrary joint compatible simulators are allowed on the enlarged output space. These results establish rank-sensitive criteria for certifying measurement incompatibility under restricted channel access.

We introduce the notions of flat translated chains of contactomorphisms and periodic flat translated chains of finite sequences of contactomorphisms, and extend to these notions the theorem of Viterbo (1992) on the multiplicity of periodic points of compactly supported Hamiltonian diffeomorphisms of $\mathbb{R}^{2n}$. More precisely, we show that every non-trivial compactly supported contactomorphism of either $\mathbb{R}^{2n+1}$ or $\mathbb{R}^{2n} \times S^1$ that is contact isotopic to the identity has infinitely many geometrically distinct flat translated chains in the interior of the support with respect to the standard contact form, that the growth-rate of such translated chains is at least linear if the contactomorphism is non-negative, as well as similar statements for periodic flat translated chains of finite sequences of contactomorphisms.

We prove that, at zero external field and for every inverse temperature $\beta>1$, the Parisi measure of the Sherrington-Kirkpatrick model is supported on an interval $[0,q_\beta]$, with a smooth density on $(0,q_\beta)$ and an atom at $q_\beta$. The primary difficulty is to exclude gaps in the support. On such a gap, the optimality conditions for the Parisi measure would force the second derivative of the associated self-consistency function $\Gamma$ to cross zero downward. We establish that it can only cross upward by showing that at a zero of $\Gamma''$, the derivative $\Gamma'''$ is a positive linear combination of two nonnegative covariances, one of which is strictly positive.

For a poset $P$, let $e(P)$ ($e^*(P)$) denote largest positive integer $k$ such that the union of the $k$ middle layers of $2^{[n]}$ does not contain a weak (strong) copy of $P$. Ellis, Ivan, and Leader showed the existence of posets $P$ for which there exists a positive real $\varepsilon_P$ such that $La(n,P)\ge (e(P)+\varepsilon_P)\binom{n}{\lfloor n/2\rfloor}$ and $La^*(n,P)\ge (e^*(P)+\varepsilon_P)\binom{n}{\lfloor n/2}$ hold, where $La(n,P)$ ($La^*(n,P)$) denotes the maximum size of a weak (strong) $P$-free family $\mathcal{F}\subseteq 2^{[n]}$. More precisely, they showed that $P=B_d$ are such posets for all $d\ge 4$, where $B_d$ is the Boolean lattice ordered by inclusion. Tompkins showed that the diamond $B_2$ is also such a poset. We apply his method to settle the case of the last Boolean poset $B_3$. We show that there exists a positive $\varepsilon$ such that $$La^*(n,B_3)\ge La(n,B_3)\ge La(n,D_6)\ge (3+\varepsilon)\binom{n}{\lfloor n/2\rfloor},$$ where $D_s$ is the poset on $s+2$ elements $a<b_1,\dots,b_s<c$. Consider the intervals $I_m=[2^{m-1}-1,2^m-2]$, $I^*_m=[\binom{m-1}{\lfloor \frac{m-1}{2}\rfloor}+1,\binom{m}{\lfloor \frac{m}{2}\rfloor}]$. It is known that for values $s$ in the major initial parts of $I_m$ and $I_m^*$, one has $La(n,D_s)=(m+o(1))\binom{n}{\lfloor \frac{n}{2}\rfloor}$ and $La^*(n,D_s)=(m+o(1))\binom{n}{\lfloor \frac{n}{2}\rfloor}$. The above equalities do not hold for the largest elements of the intervals, thus there exist $s_m\in I_m, s^*_m\in I^*_m$ such that for $s\in I_m$ we have $La(n,D_s)=(m+o(1))\binom{n}{\lfloor \frac{n}{2}\rfloor}$ if and only if $s<s_m$ and for $s\in I^*_m$ we have $La^*(n,D_s)=(m+o(1))\binom{n}{\lfloor \frac{n}{2}\rfloor}$ if and only if $s<s^*_m$. Modifying previous constructions, we obtain upper bounds on $s_m$ and $s^*_m$.

For decades, static solution concepts (Nash, Correlated, and Coarse Correlated Equilibria) and the Price of Anarchy (PoA) have formed the bedrock of algorithmic game theory, with no-regret learning proving fast convergence to such game-theoretic equilibria. We show that reducing multi-agent learning to static equilibrium and black-box regret analysis obscures underlying dynamic disequilibrium and game theoretic bounds. First, interior Nash equilibria lack $C^1$ vector field information, meaning agents cannot distinguish aligned from strictly opposing incentives. Inheriting this geometry, the worst-case pure Nash equilibria dictating robust PoA bounds manifest as topologically unstable strict saddles, and in canonical congestion games, as global repellers supported on almost everywhere strictly dominated strategies. Anchoring efficiency guarantees to these unstable states causes algebraic sensitivity; we prove that accommodating all strictly positive affine costs renders the PoA unbounded. Furthermore, projecting learning trajectories onto the discrete simplex of correlated play systematically accommodates non-rationalizable behavior. Evaluating dynamics via Coarse Correlated Equilibria or proximal refinements fails to preclude strictly dominated strategies. Moreover, optimal $O(1/T)$ swap-regret minimization does not preclude macroscopic turbulence, manifesting as chaotic limit sets even in minimal games. Finally, we examine the non-atomic limit of congestion games. Though considered highly stable with tight sub-linear $\Theta(p/\ln p)$ PoA bounds (where $p$ is the polynomial degree), we prove that under discrete-time learning, the unique equilibrium destabilizes into Li-Yorke chaos and global attractors whose time-averaged inefficiency degrades exponentially as $2^p$. These results necessitate re-evaluating worst-case equilibrium frameworks for dynamically grounded metrics.

In this paper, we present a unified construction of quantum affine vertex algebras associated to double Yangians, untwisted quantum affinization algebras, and twisted quantum affinization algebras. We show that these $\hbar$-adic quantum vertex algebras contain a dense quantum vertex subalgebra equipped with a increasing filtration. Moreover, we prove that the associated graded algebras with respect to these filtrations are isomorphic to a $\mathbb Z$-graded dense quantum vertex subalgebra of the quantum affine vertex algebras associated to double Yangians.

Given any irrational number, we construct a Denjoy example of class $C^1$ with this rotation number, that is exactly affine on each interval of the complement to the exceptional minimal set.

For $d \geq 2$, $p \geq 1$ and $\epsilon > 0$, let $N_p(d,\epsilon)$ be the smallest integer $N$ such that every $d$-dimensional subspace of $L^p[0,1]$ admits a linear embedding into $\ell_p^N$ with distortion at most $1 + \epsilon$. For fixed $d\geq 2$ and $p\geq 1$, the bound \[ N_p(d,\epsilon) \lesssim_{d,p} \epsilon^{-2(d-1)/(d+2p)} \] is established. For $p \notin 2\mathbb{Z}$, this matches the known lower bound up to constant factors. For odd integers $p$, previous upper bounds with this exponent incurred additional logarithmic factors, except in the logarithm-free case $p = 1$; for non-integral $p$, no upper bound with this exponent was previously known. For even integers $p$, isometric embeddings of dimension independent of $\epsilon$ are known. For $p \notin 2\mathbb{Z}$, the proof approximates $|t|^p$ by a polynomial with a remainder of small total variation. The polynomial part contributes no error, while the error from the remainder is controlled by an integrated equatorial-band discrepancy estimate.

Knot polynomials colored by typical representations of Lie superalgebras of type I (except $\mathfrak{psl}(n|n)$) have two variables $q$ and $t$, the latter corresponding to the complex-valued weight of the distinguished odd root. We prove that for every typical representation of a Lie superalgebra of type I, the $t$-degree of the knot polynomial is at most the number of odd roots times the genus of the knot. A complimentary bound being at least the number of odd roots times degree of the Alexander polynomial can be obtained from a specialization at $q=1$. These two bounds become equalities when the Alexander polynomial detects the genus of the knot, as is the case for alternating knots and fibered knots.

We prove global well-posedness and asymptotic stability for the three-dimensional Peskin problem, which models a closed, elastic membrane immersed in an incompressible Stokes fluid. We work with initial data in the optimal regularity space $W^{1,\infty}(\mathbb{S}^2)$, which may contain infinitely many corners. These initial configurations are instantly desingularized by the flow's parabolic smoothing effect, becoming smooth for all $t > 0$. Then we establish that the solutions converge exponentially in the $C^1$ topology to a translated and dilated conformal sphere. The stability is achieved by combining our nonlinear estimates with an exact structural decoupling of the 10-dimensional manifold of conformal steady states, demonstrating that the infinite-dimensional dissipative perturbation is strictly controlled. The core of our analysis is a functional framework on the sphere $\mathbb{S}^2$ that uses spectral Littlewood-Paley projections to control the highly singular multilinear operators arising from the fluid nonlinearity

We study the interaction between reduction rules and upper-bound functions for the Maximum Clique Problem (MCP). We show how MCP upper-bound functions can strengthen classical core and truss reductions by replacing local size conditions with upper-bound tests. This leads to the \((k,\omega^u)\)-core, the \((k,\omega^u)\)-truss, and the more general \((k,d,\omega^u)\)-truss, where the parameter \(d\) controls the trade-off between stronger reductions and additional computational cost. For each of these notions, we prove clique-preservation properties, correctness of the corresponding peeling algorithm, and running-time bounds. Based on these reductions, we introduce a general framework for improving upper-bound values for MCP. We give two concrete instantiations of the framework: one that uses only the combined truss and core reductions, and one that combines the truss and core reductions with repeated applications of structions. Computational experiments on 73 benchmark graphs show that the proposed reductions can substantially improve several standard upper-bound functions and that combining multiple reduction methods can be beneficial in practice. In particular, the combination of structions, truss and core reductions with a DSatur-based bound often reached SDP-level upper-bound values faster than direct SDP computation; on the tested graphs with edge density below \(0.7\), it did so in every case. Using the truss and core reduction with the Lovász theta upper-bound function, we also improve the previously best certified integer upper-bound values for three difficult DIMACS instances whose exact clique numbers are not known. In particular, we improve upper-bound values for graph C500.9 from 83 to 73, for graph C1000.9 from 122 to 115, and for graph C2000.9 from 177 to 168.

Prefabricated prefinished volumetric construction (PPVC) builds room-sized modules in factories that operate as flexible job shops. A module waits 12 to 48 h after casting, waterproofing, and painting, while its station is free for other modules. Existing learned dispatchers release the next operation before this wait ends, so their plans need repair. This paper presents SLAB (Scheduling under time-Lags with an Admissible Bound), a decision process for dispatchers trained by deep reinforcement learning. Its transition releases a module only after its wait, and its reward is the decrease of a proven lower bound that includes the waits. Training on 10-module orders with halved waits also teaches the policy the station contention of larger orders. On three existing networks, repaired plans are 42% to 133% longer than the same network's schedules inside SLAB. On orders generated from Singapore's PPVC guidebook, one trained policy gives schedules 2.2% to 8.1% shorter than the shortest-processing-time rule at 10 to 80 modules. At 50 and 80 modules, it outperforms a time-lag memetic algorithm by 6.7% and 7.4% and constraint-programming search started from that rule's schedule and given 600 s by 6.6% and 7.6%. The memetic algorithm gives shorter schedules on orders of 10 modules, and the same search given 300 s on orders of 10 and 20 modules. A module factory can therefore train one dispatcher on small orders and run it without a solver on production batches of 50 to 80 modules.

We establish a new Bernstein-type deviation inequality for classes of functions whose complexity is characterized through subgraphs. The inequality is non-asymptotic, involves explicit constants, and features a relative normalization by the probability level. Applied to kernel density estimation, it produces a location and bandwidth-adaptive error bound between the estimator and the smoothed density, holding simultaneously over all points on the real line and all positive bandwidths. The proof is elementary, combining a new symmetrization principle, which incorporates the relative normalization, with the maximal sub-Gaussian inequality, and requires neither concentration nor entropy-integral arguments. When specialized to classes of sets, our technique improves the constants in the classical Vapnik-Chervonenkis inequality with relative deviation of Anthony and Shawe-Taylor (1993), reducing the factor 4 S A (2n) to S A (2n) in the right-tail inequality and to 3 S A (2n) in the left-tail inequality.

Solving partial differential equations (PDEs) in high dimensions remains challenging due to the curse of dimensionality. We propose a neural-network-based framework that reformulates PDEs as inf–sup optimization problems through the introduction of a Lagrange multiplier. The primal solution and the associated Lagrange multiplier are parameterized by two networks and are computed via an iterative saddle-point optimization procedure. We prove the theoretical equivalence between the proposed optimization formulation and the original PDE problem, and we derive rigorous error estimates that quantify the total approximation error in terms of the network approximation error, statistical (sampling) error, and optimization error. Numerical experiments demonstrate the accuracy, stability, and efficiency of the proposed method for solving high-dimensional PDEs.

We study a general decision-dependent contextual stochastic program (DD-CSP) in which uncertainty depends on both exogenous contextual information and endogenous decisions. To learn the potentially complex dependence of uncertainty on decisions and contextual information, we employ several nonparametric regression models, including k nearest neighbors (kNN), classification and regression trees (CART), and ReLU neural networks. To account for estimation errors in predicting the uncertainty, we adopt an empirical residuals-based decision-dependent sample average approximation (ER-DD-SAA) framework, which adds empirical residuals to the point predictions from the learned regression models. For each nonparametric regression model, we develop exact mixed-integer programming (MIP) representations that can be seamlessly embedded within the ER-DD-SAA framework. For two-stage ER-DD-SAA problems with kNN, we further propose a tailored decomposition algorithm, named BD-CG, that combines Bender's decomposition with constraint generation. Under suitable assumptions, we prove that the proposed BD-CG converges to a global optimum within a finite number of iterations. From a statistical perspective, we establish the consistency and asymptotic optimality of ER-DD-SAA with all three nonparametric regression models under mild regularity conditions. Numerical experiments on a newsvendor problem with pricing and a two-stage facility location problem demonstrate that the ER-DD-SAA model with nonparametric learning consistently outperforms a parametric benchmark in out-of-sample performance and the proposed reformulations and algorithm substantially improve computational tractability.

We introduce TITO_Explore, a software package for representing and computing with Translation-Invariant Total Orders (TITOs). We define a canonical window notation for TITOs and design and implement algorithms for several computational tasks involving them. The package normalizes the window notation of a given TITO into its canonical form, computes its inversion set, compares the weak order between two TITOs, and computes the join of two specified TITOs. Our weak order comparison algorithm operates by partitioning the inversion sets into disjoint subsets, thereby breaking down the comparison problem into evaluations of paired subsets. The join algorithm uses an edge-weighted directed graph to represent inversions and converts the problem of finding the join into a weighted path problem in the graph.

Boson sampling demonstrates quantum advantage through the interference of indistinguishable particles, with output probabilities governed by matrix permanents. Realizing it on deterministic, matter-based platforms requires encoding the bosonic modes in finite-dimensional local Hilbert spaces, which introduces a leakage channel absent in linear optics: multi-particle bunching beyond the local truncation $d$. We develop a unified framework for non-interacting sampling on the irreducible representations of compact Lie groups, in which the transition amplitude is the immanant of a submatrix of the single-particle transition matrix, recovering the permanent in the bosonic case. Within this framework we bound the bunching leakage through a Dyson-series analysis: decomposing the correlated many-body leakage operator into independent random matrices and applying non-commutative concentration inequalities, we prove, in a Gaussian model of the transition matrix, that its spectral norm concentrates at $\tilde{O}(\sqrt{n})$ rather than the $O(n)$ worst-case of prior spin-based emulations; the passage to the physical Haar ensemble is reduced to a single submatrix-comparison input, verified at leading order. Exact numerics across local dimensions $d=2$–$5$ indicate that the bound is tight, the Haar-ensemble norm matching the closed form $\sqrt{d(n-d+1)}$ to sub-percent accuracy. This tightens the required mode number from $m=\Omega(n^4)$ to the near-optimal $m=\tilde{\Omega}(n^{1+2/(d-1)})$; for a spin-1 representation ($d=3$) the overhead falls to $m=\tilde{\Omega}(n^2)$, matching the collision-free threshold. The result is independent of particle statistics and applies across finite-dimensional Lie-symmetric architectures, quantifying the spatial resources needed to preserve sampling hardness.

This paper investigates the dependence of the control cost for a wave equation with respect to perturbation by a time-independent potential $\lmbd V$ scaled by a large parameter $\lmbd$ on a compact Riemannian manifold. We introduce the geometric control condition~\eqref{GCC+}, a variant of the geometric control condition of Bardos–Lebeau–Rauch–Taylor, tailored to accommodate the influence of the potential $V$. We show that~\eqref{GCC+} is necessary and sufficient for the existence of a uniform observability cost with respect to the large parameter $\lmbd$. We provide geometric examples satisfying~\eqref{GCC+} and estimate the blow-up rate of the observability cost in situations where it fails. The proofs rely on semiclassical and second microlocal defect measures.

Let $P_n$ denote a path on $n$ vertices. A simple finite graph $G$ is called $P_n$-irregular if any two distinct vertices of $G$ belong to a different number of subgraphs of $G$ isomorphic to $P_n$. Alternatively, for a fixed vertex $r$ of $P_n$ (the root), $G$ is called $(P_n)_r$-irregular if any two distinct vertices of $G$ act as the root $r$ in a different number of subgraphs of $G$ isomorphic to $P_n$. This paper proves that for each integer $k \geq 4$, there exists an infinite family of graphs that are simultaneously $P_n$-irregular and $(P_n)_r$-irregular for every integer $n$ satisfying $4 \leq n \leq k$ and every root $r$ of $P_n$. For the path $P_3$, we observe that no nontrivial $(P_3)_r$-irregular graphs exist if $r$ is the central vertex. In contrast, if $r$ is an end-vertex of $P_3$, an infinite collection of graphs is constructed that are both $P_3$-irregular and $(P_3)_r$-irregular. In particular, these results confirm the Strong Conjecture about $F$-irregular graphs for the case where $F$ is a path $P_n$.

We give counterexamples to two optimization problems in dimensions eight and nine. 1. The Cartesian-product problem posed by Cassels for critical determinants and later formulated by Zong for lattice packings and for packings allowing translations but not rotations: whether the corresponding product inequalities are always equalities. The counterexample also disproves the natural product formula for unrestricted congruent packings. 2. A question raised by Sarnak and formulated as a conjecture in Chiu: whether, among unit-volume flat tori, height is minimized by a lattice maximizing the length of its shortest nonzero vector.

We study the $(p, q)$-deformed generalized Weyl algebra generated by variables $X, Y$ and $Z_p$ satisfying the $(p, q)$-commutation relations $XY-qYX=h Y^sZ_{p}, XZ_p=pZ_pX$, and $Z_pY=pYZ_p$, with $s\in \mathbb{N}_0$. Within this framework, we investigate the noncommutative binomial formula $(X+Y)^n$ and related identities. In particular, we show how the associated normal ordering coefficients can be expressed in terms of $(p,q)$-deformed $s$-rook numbers. We treat several special cases explicitly, recovering known results from literature as well as deriving new ones.

Let $r$ and $n$ be two positive integers and $S=K[x_1,\ldots,x_{n+r-1}]$, the ring of polynomials in $n+r-1$ variable, over a field $K$. We consider the squarefree monomial ideal $I_{n,r}:= x_1 \cdots x_{r-1} (x_{r},\ldots,x_{r+n-1}) \subset S$ and we prove several results regarding the Hilbert depth of $S/I_{n,r}$. Also, we consider the special case $n=r$.