Papers

We analyze fractional phase-transition energies with periodic hetero\-genei\-ties at the critical $H^{1/2}$ scaling. We prove that the $\Gamma$-limit is a sharp-interface functional whose surface energy density combines homogenization and averaging effects. The resulting coefficient is a weighted combination of the minimum and the mean of the oscillatory parameter, reflecting the coexistence of multiple interaction scales. This behavior is specific to the critical regime and differs from the case $s>1/2$.

Let ${\bf x} = (a_i)_{i \geq 0}$ be an infinite word over a finite alphabet $\Sigma$. Let $\rho (n)$ be the factor complexity function for $\bf x$ and ${\rm Pal}(n)$ be the palindrome complexity function for $\bf x$. We give a new relationship between these two quantities; namely, if $\bf x$ is not ultimately periodic, then $$ \lim_{n \rightarrow \infty} {{ {\rm Pal} (n) \log ({\rm Pal} (n) + 1)} \over {\rho (n)}} = 0. $$ Furthermore, we prove that the numerator in this result is essentially optimal.

In this paper, we show that generalized Chebyshev polynomials of the second kind, associated with the root system $A_2$, are not arithmetically exceptional. We achieve this by studying the norms of certain cyclotomic elements that admit a parametrization of finite fields.

We study the typical-code (quenched) behavior of the false-alarm (FA) and missed-detection (MD) error exponents of the Neyman-Pearson test associated with soft covering, complementing the average-code (annealed) analysis that has been carried out in a companion paper [1]. We prove that, as the block-length tends to infinity, for almost every randomly selected fixed-composition codebook, the negative normalized logarithms of both error probabilities converge to their respective average-code exponents. In other words, the error exponents are self-averaging. We then extend the scope and study a mismatched likelihood ratio test that assumes the wrong channel model. Here, we derive the mismatched error exponents, show that self-averaging persists under mismatch, and characterize the degradation. In particular, we characterize the coding rate beyond which the two kinds of error exponents cannot be positive at the same time, which in the matched case, is given by the channel input-output mutual information rate.

In this paper, we study the nested Hilbert scheme $(\mathbb{A}^2)^{[n,n+1]}=\mathrm{Hilb}^{n,n+1}(\mathbb{A}^2)$ from a combination of deformation theory, torus actions, and Young diagram combinatorics. We first recall the scheme theory and functor basics needed to define Hilbert schemes. We then use a classic result on first-order deformations to identify $T_I(\mathbb{A}^2)^{[n]}\cong \mathrm{Hom}_{\mathbb{C}[x,y]}(I,\mathbb{C}[x,y]/I)$. For a nested pair $I\subset J$, with $\dim_{\mathbb{C}}\mathbb{C}[x,y]/I=n+1$ and $\dim_{\mathbb{C}}\mathbb{C}[x,y]/J=n$, the tangent space becomes a compatibility kernel $T_{(I,J)}(\mathbb{A}^2)^{[n,n+1]}\cong \ker(\mathrm{Hom}(I,R/I)\oplus \mathrm{Hom}(J,R/J)\to \mathrm{Hom}(I,R/J))$. The torus-fixed points are indexed by a partition $\lambda\vdash n+1$ together with a removable corner $c$ of its Young diagram. This corner is not only combinatorial, but also the monomial form of a one dimensional socle direction in $R/I_\lambda$. The blow-up map to $(\mathbb{A}^2)^{[n]}\times \mathbb{A}^2$ has fibres given by projective spaces of one-dimensional quotients of $J/\mathfrak m_pJ$, whose torus-fixed points are addable boxes of the smaller diagram. These two local fibres explain how the universal family, the blow-up geometry, and Young diagram combinatorics come together in the study of the local geometry of the nested Hilbert scheme of points. Finally, we derive the tangent weight formula at a fixed point $(I_\lambda,I_{\lambda\setminus c})$ in the torus convention used in the paper. Using the standard arrow basis, we show in the proof how the arm-leg weights are modified by the compatibility kernel through a shortening rule determined by $c$. A Macaulay2 verification computes the compatibility kernel from monomial syzygies and checks the weight formula for all partitions of size at most $16$.

This paper investigates the Markovian dynamics of open two-state quantum systems defined over the real numbers (rebits). Two main objectives are pursued. First, we present a comprehensive classification of Markovian rebit quantum channels, i.e. one-parameter semigroups of completely positive, trace-preserving (CPTP) maps acting on the rebit state space. We show that a full characterisation of their action can be achieved and that describing these channels as solutions of the GKSL equation allows us to explicitly identify the associated Lindblad generators and conditions for complete positivity. Second, we present an original application of this classification to colour perception. Using a recent model in which perceived colours arise from Lüders measurements on the rebit state space, we show how chromatic distortion induced by a non-neutral illuminant can be modelled by a Markovian rebit channel that progressively diminishes colour distinguishability. Other types of channels could be used to study colour vision deficiencies. These phenomena are illustrated by simulations on digital images, highlighting the relevance of rebit Markovian dynamics in modelling colour vision.

The paper introduces the model of a pair of branching processes $\{\boldsymbol{Z_n} = \left(Z_n^{(1)}, Z_n^{(2)}\right), \; n \in \mathbb{N}_0\}$ in a joint random environment. If the environment is fixed then the sequences $\{Z_n^{(1)}, \; n \in \mathbb{N}_0\}$ and $\{Z_n^{(2)},\; n \in \mathbb{N}_0\}$ are independent branching processes in a varying environment. This model is a particular case of a more general model of a multitype branching process in a random environment. We establish the asymptotic relation ${\bf{P}}\left(Z_n^{(1)} > 0, Z_n ^{(2)}>0 \right) \sim C n^{-a}$ as $n \to \infty$, where the parameter $a$ depends only on the correlation coefficient $\rho$ of an increment of a two-dimensional associated random walk.

A small Wasserstein distance does not certify that a transformation is admissible. In evidence-constrained, semantic, causal, physical, monotone, or risk-sensitive learning, one must ask not only how far two probability laws are, but whether mass has moved in a direction allowed by available information. We introduce conditional random ordered transport spaces (CROTS), a class of \(L^0\)-valued spaces of random probability measures equipped with a Wasserstein ambient metric, a closed stochastic order, hard and soft ordered transport discrepancies, and a conditional risk functional for evaluating order violation under an evidence sigma-field. The central object is an order-admissible transport geometry for random measure-valued dynamics, distinct from cone-valued metrics, ordered Kantorovich constructions, random Wasserstein spaces alone, and model-specific residuals for generative paths. We develop the foundations of CROTS as a space theory for reliable distributional learning. The results include well-posedness and duality for hard and soft ordered transport, soft-to-hard variational convergence, measurability and completeness of the random lifted space, reductions to classical Wasserstein and ordered geometries, ordered geodesics, constrained barycenters and projections, conditional risk-transport duality, and separation of order-violating distributions. The main stability theorem shows that random learning dynamics may converge in the ambient Wasserstein metric while its local admissibility leakage follows a separate conditional order-risk recursion. The resulting asymptotic order-risk floor provides a mathematical language for evidence overreach, ordered distribution shift, robustness failure, and admissible distributional dynamics.

A new fractional ambiguity function (NFrAF) derived from the fractional Fourier transform is introduced as a generalization of the classical ambiguity function. The fundamental analytical properties of the NFrAF, including symmetry, marginality, and Moyal type identities, are rigorously established. After verifying its ability to detect and localize monocomponent and multicomponent linear frequency modulated (LFM) signals, the NFrAF is integrated into a convolutional neural network based machine learning framework for signal classification. Owing to its superior time frequency resolution and localization, the NFrAF provides a more informative input representation than conventional methods such as the spectrogram and classical ambiguity function. Experimental results on simulated datasets demonstrate consistent improvements in classification accuracy, highlighting the effectiveness of the proposed representation for data driven signal analysis.

This paper is an expository account of cyclotomic spectra. They are spectra (in the sense of homotopy theory) with additional structure that includes an action of the circle group, which we will denote by $\mT$, for torus. Such objects come up in algebraic $K$-theory and its close relatives topological Hochschild homology $\THH$ and topological cyclic homology $\TopC$. They figure prominently in the recent disproof of the {\TC} for chromatic heights greater than 1 by Robert Burklund, Jeremy Hahn, Ishan Levy and Tomer Schlank . Those authors show that for each $n\geq 1$ and each prime $p$, there is a $p$-local ring spectrum $X$ of chromatic height $n$ such that $L_{K (n+1)}\TopC (X)$ and $L_{T (n+1)}\TopC (X)$ (see \cref{def-KT-KK}) are distinct. The present work is part of my attempt to understand theirs

We develop the theory of "automatic actions": (semi)groups acting by $\omega$-regular transformations on an $\omega$-regular language, showing that it covers a large class of heretofore-unrelated examples. We focus on the subclass of actions by "bounded" $\omega$-regular transformations, those for which the Büchi automata encoding the action do not have two connected non-trivial cycles. We show that, for bounded actions of inverse semigroups, the orbit relation is also $\omega$-regular. We deduce a number of corollaries, in particular decidability, for such actions, of minimality, topological transitivity, aperiodicity, and order of elements. More generally, every first-order statement over the space of the action, involving the action of specific semigroup elements as well as the relation "being in the same orbit", is decidable. We also apply this result to the study of Julia sets of post-critically finite polynomials, and show that the encoding of Fatou components is also computable; thus every first-order statement involving intersection, disjointness etc. of Fatou components or their full orbit under the polynomial, is decidable.

This paper presents a formulation for strain localization in softening plasticity based on a deformable Cosserat model. The approach enables the direct use of standard elastoplastic constitutive models formulated for a classical Cauchy continuum, without modifying the stress update algorithm or consistent tangent operator. A key feature of the framework is the strict separation of dissipative and energetic mechanisms: all dissipation is confined to the macro-continuum, while the micro-continuum contributes only through linear elastic terms associated with the director field. As a result, the constitutive structure of the elastoplastic model is preserved, and existing models can be employed as black-box components. The internal length scale arises naturally from the micro-continuum and governs the development, interaction and selection of localization patterns, rather than acting as a diffusive parameter. The formulation is easy to implement within standard finite element frameworks, requiring only additional linear contributions to the residual and tangent operators. The performance of the approach is assessed through benchmark problems involving shallow foundations on soil, a demanding test due to complex and unstable localization mechanisms. Both Tresca and Matsuoka-Nakai plasticity models are considered, including cases with highly unstable post-peak responses. Numerical results show convergence of load-displacement responses, dissipated energy and shear-band patterns upon mesh refinement, even in the presence of nonlinear interacting localization processes. These findings demonstrate a robust and physically consistent approach for the analysis of strain localization in softening plasticity.

For every positive integer $a$ which is coprime with $p$, $p$ is an odd prime, we denote by $\overline{a}$ the unique integer satisfying $1\leq \overline{a}\leq p$ and $a\overline{a}\equiv 1(\mathrm{mod}~p)$. Put $$L(p)=\{a\in Z^+:(a,p)=1,2\nmid a+\overline{a}\}.$$ The elements of $L(p)$ are called D.H. Lehmer numbers. The main purpose of this paper is to prove that for any fixed odd prime $p$, every sufficiently large number unless it is congruent to 15 or 16$(\mathrm{mod}~{16})$ is representable as the sum of 14 fourth powers of D.H. Lehmer numbers. Furthermore, every sufficiently large number is representable as the sum of 16 fourth powers of D.H. Lehmer numbers.

In this paper, we derive monotone quantities for harmonic functions on asymptotically flat 3-manifolds with simple topology and nonnegative scalar curvature. These monotone quantities are constant on spatial Schwarzschild manifolds outside rotationally symmetric spheres. To derive monotone quantities, our method is different from the ODE analysis in Xia-Yin-Zhou \cite{Xia} and Mazurowski-Yao \cite{Maz}, we follow the strategy developed in Miao \cite{Miao}. As applications, we recover and generalize some geometric inequalities and mass-capacity inequalities in Miao \cite{Miao} and Oronzio \cite{Oronzio}. Furthermore, we obtain the integral identities for the mass-capacity ratio which is parallel to the results in Miao \cite{Miao}.

We prove that the order of vanishing of a natural twist of the $P$-adic Carlitz zeta values at the positive "even" integers is always $1$. We also obtain the same result for $P$-adic Pellarin $L$-series, and for almost all $P$-adic Dirichlet-Goss $L$-series. To do so, we demonstrate a $P$-adic version of a reduced variant of the class formula for Anderson $t$-modules proven by Anglès, Ngo Dac and Tavares-Ribeiro in 2020.

The main goal in regression modelling consists in approximating the conditional mean of a response given a set of features. A regression function is said to be calibrated if the resulting mean estimates match the true conditional means for almost every set of features. Aiming for calibration seems not achievable in practice as one typically deals with finite samples of noisy observations. A weaker notion of calibration is auto-calibration, and it means that the expectation of responses being given the same mean estimate matches this estimate. This notion is important, e.g., in insurance pricing as it ensures no cross-subsidization between different price cohorts. In this paper, we show that boosting trees can be used to test necessary conditions for calibration and auto-calibration, respectively. The practical relevance of our approach is supported by a numerical example, in which the proposed tests prove to be very powerful on a large insurance dataset.

Weyl conformal geometry is the natural underlying geometry of gauge theories of Weyl group (of dilatations and Poincaré symmetry), such as Weyl quadratic gravity and its generalisation, Weyl-Dirac-Born-Infeld action (WDBI). These are local, Weyl-anomaly free (quantum) gauge theories of gravity. We describe Weyl gauge symmetry from a more familiar Riemannian view of Weyl gauge invariant dressed fields by the Wilson line of dilatations. Weyl geometry can then be seen as Riemannian geometry of non-local dressed metric ($g_{\mu\nu}^*$), at the "cost" of (gauge-induced) non-commutativity in the UV, due to Wilson line. Then Weyl quadratic gravity and WDBI actions of Weyl geometry, which are Weyl gauge invariant in $d$ dimensions, have the same expression in Riemannian geometry defined by $g^*_{\mu\nu}$. This is a non-local map and dual description of the two geometries and actions in the symmetric phase. Unlike for the metric, the equation of motion of Weyl gauge field ($\omega_\mu$) does not commute with the dressing of the metric. Quantum non-locality, in particular entanglement, and non-commutativity are (gauge-invariant) physical artefacts of "translating" (local) Weyl geometry and Weyl gauge covariance into our real-world Riemannian geometry of Weyl gauge invariant observables, and they are evidence of Weyl gauge symmetry. At lower energies, $\omega_\mu$ becomes massive, can decouple and Einstein-Hilbert action and commutativity are recovered. The case of a light $\omega_\mu$ is also discussed.

Let $a,b$, and $n$ be positive integers and let $S=\{x_1, \cdots, x_n\}$ be a set of $n$ distinct positive integers. We denote by $[S^a]$ the $n\times n$ matrix having the $a$th power of the least common multiple of $x_i$ and $x_j$ as its $(i,j)$-entry. For $x\in S$, let $G_{S}(x)=\{d\in S: d<x, d|x \ {\rm and} \ (d|y|x, y\in S)\Rightarrow y\in \{d,x\}\}$. In this article, we show that $[S^a]$ divides $[S^b]$ in the ring of $n\times n$ matrices over the integers if $a|b$ and $S$ is gcd closed (i.e., $\gcd(x_i,x_j)\in S$ for all integers $i$ and $j$ with $1\le i,j\le n$) and satisfies the condition $\mathcal G$ (that is, for any $x\in S$, either $G_S(x)$ contains at most one element, or $G_S(x)$ contains at least two elements and satisfies that ${\rm lcm}(y_1,y_2)=x$ and $\gcd(y_1,y_2)\in G_S(y_1)\cap G_S(y_2)$ for any $\{y_1,y_2\}\subseteq G_S(x)$). This confirms a conjecture of Hong proposed in (2026, Bull. Aust. Math. Soc., 113, 231-243).

For real algebraic curves in the plane with singularities we investigate the relation between normals and double normals and the critical points of the squared distance function (up to topological equivalence). For the distance to a given point we show that the topological discriminant consists of the (traditional) evolute, together with some distinguished normal lines at algebraic singular points. We pay special attention to the behavior at points, where the curve is smooth, but only $C^1$ embedded. We discus the differences and similarities with the ED-discriminant in the complex theory. We give counting formula's for normals and double normals, relating them with maxima and minima of the distance function.

In the category of topological Abelian groups, we consider the usual notion of an extension $E=(B \to X \to A)$ of $B$ by $A$, together with the notion of a weakly split extension, i.e., an extension for which the projection $X \to A$ admits a continuous section $A \to X$. Given a weakly split extension $E$, the topological Abelian group $X$ is homeomorphic to $B \times A$, although in general it is not algebraically isomorphic to $B \times A$. For two topological Abelian groups $A$ and $B$, we study the Abelian group $E^{\mathrm{ws}}_{\mathrm{TA}}(A,B)$ of weakly split extensions of $B$ by $A$, modulo extension isomorphisms. We show that $E^{\mathrm{ws}}_{\mathrm{TA}}(A,B)$ can be described as the group of all continuous sum structures defined on the product space $B \times A$ (up to topological isomorphism), with $B$ as a topological subgroup and $A$ as a topological quotient. We also provide an alternative description of $E^{\mathrm{ws}}_{\mathrm{TA}}(A,B)$ as a quotient $Z_c(A,B)/B_c(A,B)$, where $Z_c(A,B)$ consists of cocycles given by continuous maps $A \times A \to B$, and $B_c(A,B)$ denotes the corresponding coboundaries. Furthermore, we compare $E^{\mathrm{ws}}_{\mathrm{TA}}(A,B)$ with the group of standard extensions $E_A(A,B)$, where $A$ and $B$ denote the underlying Abelian groups, and relate these constructions by means of a six-term exact sequence. Although the Bohr topology of discrete Abelian groups has been investigated by many workers, there still remain many parts that are not well understood. Here, as an application of the methods developed in the paper, new examples of nontrivial $ws$-extensions for discrete Abelian groups equipped with the Bohr topology are provided and some related open questions are also proposed.

Stimulated by Aldroubi and his collaborator's recent work on dynamical sampling, we consider a homogeneous discrete dynamical system of the form $f_n = A f_{n-1}= A^{n}f, \quad f_0 = f$, where A is a bounded linear operator on a separable Hilbert space H, which is known as the evolution operator, and $f_0 \in \mathcal{H}$ is the unknown initial state. The associated dynamical samples are given by the collection $\{\langle A^n f, g\rangle: g \in \mathcal{G, 0 \leq n < n < L(g)}\}$, where $\mathcal{G} \subset H$, is a finite or countable sampling set and $L$ is a function $L: G \rightarrow \mathbb{N} \bigcup \{\infty\}$. We analyze the stability of perturbed dynamical sampling systems in the sense of Nagy and Pollard-Hilding. More precisely, we establish sufficient conditions for the stable recovery of an initial state from perturbed dynamical samples obtained by changing the sampling vector, the evolution operator, ro simultaneously both, within the framework of Nagy-type and Pollard-Hilding-type perturbation of frames.

We study the second largest spectral gap from $1$ of the normalized Laplacian of a graph, a quantity that appears in the literature in connection with random walks, expander graphs, and Ramanujan graphs. We relate it to the classical Cheeger and dual Cheeger constants, and we introduce a new Cheeger-type constant admitting a probabilistic interpretation in terms of two-step random walks. For this constant, we establish sharp inequalities analogous to the classical Cheeger inequalities.

Building upon the algebraic framework of trace monoids introduced by Giscard and Rochet, we establish a new determinant formula for the Ihara zeta functions of certain digraphs. We present our results through two main theorems. First, at the algebraic level, we show a general determinant formula expressed by a Cayley determinant over the hike monoid ring, which provides a unifying perspective on several known determinant expressions. Second, by evaluating the lengths of hikes through a natural ring homomorphism, we prove that the reciprocal of the Ihara zeta function can be explicitly expressed as a determinant whose dimension is equal to the number of simple cycles of the line digraph without backtrack.

In this paper, we introduce and study the notion of weakly Q-ideals in commutative rings.

In 2011, Csikvári [Electron. J. Combin. 18 (2011), $\#$P182] proved that among all graphs with a prescribed number of edges, the largest matching root is attained by a threshold graph, and conjectured that the extremal graph should be `as star-like as possible.' In this paper, we give a complete and affirmative answer to this problem and extend it to the setting of uniform hypergraphs. We prove that for every $k$-graph $\mathcal{H}$ with $m$ edges, its largest matching root satisfies $$\lambda(\mathcal{H})\le m^{1/k},$$ with equality if and only if $\mathcal{H}$ is intersecting. For $k=2$, after deleting all isolated vertices, the resulting graph must be the star $K_{1,m}$ or a triangle, thereby confirming Csikvári's conjecture. Moreover, if the matching number $\nu(\mathcal{H})\ge 2$, then \[ \lambda(\mathcal{H})\le \left(\frac{m+\sqrt{m^2-4(\nu(\mathcal{H})-1)}}{2}\right)^{1/k}, \] with equality if and only if $\nu(\mathcal{H})=2$ and $\mathcal{H}$ has exactly one $2$-matching.

In this work, we propose an operator-splitting variational quantum algorithm, termed Dirac-sVQA, for simulating the nonlinear Dirac equation (NLDE). The main difficulty arises from the state-dependent nonlinear interaction, its time-discrete update depends explicitly on the intermediate spinor state and, in general, cannot be implemented as a fixed state-independent unitary circuit. To address this difficulty, we decompose the NLDE evolution into a structured linear Dirac substep and a nonlinear variational correction. The linear substep is implemented by a spinor-Fourier Dirac propagator on a joint position-spin register, preserving the spin-momentum coupling and mass-induced spin evolution of the Dirac operator. The nonlinear correction is reformulated as a measurement-based variational update through a small set of overlap, self-channel, and cross-channel observables. We provide the corresponding quantum circuits and derive measurement-aware resource and complexity estimates. Numerical experiments in several nonlinear regimes show that Dirac-sVQA accurately captures both the total density and the componentwise spinor dynamics, agrees well with classical Fourier pseudospectral splitting solutions, and exhibits stable error behavior over time. These results provide numerical evidence for the feasibility of operator-splitting variational quantum simulation for nonlinear relativistic wave equations.

A classical theorem of Boshernitzan states that if $f$ is a function which belongs to a Hardy field and which satisfies $|f(x)|\prec x^{\ell}$ for some $\ell\in \mathbb{N}$, then the sequence $(f(n))_{n\in \mathbb{N}}$ is uniformly distributed modulo 1 if and only if $\lim_{x\to\infty}\frac{|f(x)-p(x)|}{\log(x)} = \infty$ for all $p(x)\in \mathbb{Q}[x]$. We provide a new proof of this result using methods from summability theory and we extend Boshernitzan's criterion by obtaining necessary and sufficient conditions for $f$ to be uniformly distributed modulo 1 with respect to a broad class of weighted averages. As an application of our results, we show that for the function $f(x) = x^{3/2}$ and for any $(a,b)\subset [0,1]$, and all sufficiently large $N\in\mathbb{N}$, there is an $n\in [N-N^{\frac{1}{4}},N]$ such that $f(n)\mod 1\in (a,b)$.

We investigate the monogenity of irreducible factors of the Fibonacci polynomials $F_n(x)$ and the Lucas polynomials $L_n(x)$. Our main results show that for every odd positive integer $n$, all irreducible factors of $F_n(x)$ are monogenic, and for every even positive integer $n$, all irreducible factors of $L_n(x)$ are monogenic.

Practical distributed deployments inherently suffer from asynchronous signal arrivals, which exacerbate multi-user interference and degrade system performance, especially for coherent transmission. To natively mitigate the asynchronous reception effect, this paper proposes integrating fluid antenna systems (FASs) into distributed cell-free massive MIMO systems, exploiting their reconfigurable spatial positions to release additional spatial degrees of freedom (DoFs). We establish the FAS-enabled data transmission model with asynchronous reception, i.e., delay phases. We also derive the analytical downlink spectral efficiency (SE) performance of the proposed system under coherent and non-coherent transmissions, using low-complexity Maximum Ratio (MR) precoding to provide fundamental theoretical bounds. Specifically, we propose a novel nonmonotone accelerated projected gradient ascent algorithm to jointly optimize FAS positions and power control coefficients, maximizing the downlink sum SE. Numerical results demonstrate that while asynchronous reception severely degrades system performance for coherent transmission, the spatial DoFs unlocked by optimized FAS positions, along with efficient power control, can significantly counteract the effects of unknown delay phases and outperform traditional fixed-position antennas. For non-coherent transmission, which inherently bypasses asynchronous reception, the application of FAS leverages spatial reconfigurability to natively maximize signal strength and achieve more pronounced SE gains. Ultimately, our proposed FAS-enabled system, coupled with efficient power control, mitigates performance degradation due to asynchronous reception and outperforms traditional fixed-position antennas, paving the way for the practical deployment of FASs in robust, highly efficient 6G cell-free massive MIMO systems.

Let $R = k[x_1,\ldots,x_n]$ be a polynomial ring in $n$ variables over a field $k$, and let $I$ be a monomial ideal of $R$. If $I$ is an almost complete intersection, then we provide an explicit formula for the Castelnuovo-Mumford regularity of $I$ in terms of the powers of the dominant variables appearing in the regular sequence contained in $G(I)$ of length $|G(I)|-1$, where $G(I)$ is the set of minimal monomial generators of $I$. Furthermore, if $I$ is a dominant ideal or an almost complete intersection ideal, then we show that $\operatorname{reg}(\overline{I}) \leq \operatorname{reg}(I),$ where $\overline{I}$ denotes the integral closure of $I$. This provides a positive answer to the Küronya-Pintye conjecture for these two classes of monomial ideals. In addition, we give some examples to clarify these results.