Let $(R,\mathfrak m)$ be a Cohen-Macaulay local ring of dimension $d\geq 2$ and $I$ an $\mathfrak m$-primary ideal. Let rd$(I)$ be the reduction number of $I$ and n$(I)$ the postulation number. We prove that for $d=2,$ if n$(I)=ρ(I)-1,$ then rd$(I) \leq$n$(I)+2$ and if n$(I)\neq ρ(I)-1,$ then rd$(I)\geq$n$(I)+2.$ For $d \geq 3$, if $I$ is integrally closed, depth gr$(I) = d-2$ and n$(I)=-(d-3).$ Then we prove that rd$(I)\geq$n$(I)+d$. Our main result is to generalize a result of T. Marley on the relation between the Hilbert-Samuel function and the Hilbert-Samuel polynomial by relaxing the condition on the depth of the associated graded ring with the good behaviour of the Ratliff-Rush filtration with respect to $I$ mod a superficial element. From this result, it follows that for a Cohen-Macaulay ring of dimension $d\geq2$, if $P_{I}(k)=H_{I}(k)$ for some $k \geq ρ(I)$, then $P_{I}(n)=H_{I}(n)$ for all $n \geq k.$

This paper presents a novel technique to build squint-free massive phased arrays. This is accomplished by explicitly implementing a spatial IDFT to cancel out the DFT imposed by the array nature which causes beam squint. In addition, the paper analyzes the beam-squint issue, which arises from two mechanisms: the coherent bandwidth limitations and the systematic delay spread in the array. These mechanisms reduce the signal-to-noise ratio and cause inter-symbol interference. This work also highlights the importance of utilizing OFDM modulation to enhance signal quality by mitigating the self-interference issue. A numerical solver is used to simulate and verify the IDFT squint-free implementation and to estimate the signal quality limitations in massive arrays.

The theory of point-particles in classical electrodynamics has a well-known problem of infinite self-energy, and the same is true of quantum electrodynamics. Instead of concluding that there is no such thing as a true point-particle, it is shown here how to remove the infinities by supposing that the electromagnetic field tensor has a symmetric part. This does not change the physics, as the equation of motion and the antisymmetric part of the retarded fields appearing in the equation of motion are unaffected. The symmetric part of the field tensor is not observable and therefore it need not be gauge-invariant, whereas the antisymmetric part is observable, gauge-invariant, and satisfies both the Maxwell Equations and the field equations governing the whole field tensor. This approach goes well beyond prior efforts at classical renormalization, and also entails a new derivation of the Lorentz-Abraham-Dirac (LAD) equation of motion. Implications related to General Relativity are described in the Appendix.

In Luparello et al. 2023, a new and hitherto unknown CMB foreground was detected. A systematic decrease in Cosmic Microwave Background (CMB) temperatures around nearby large spiral galaxies points to an unknown interaction with CMB photons in a sphere up to several projected Mpc around these galaxies. We investigate to which extent this foreground may impact the CMB fluctuations map and create the so-called CMB anomalies. Using the observed temperature decrements around the galaxies, and making some general assumptions about the unknown interaction, we propose a common radial temperature profile. By assigning this profile to nearby galaxies in the redshift range $z=[0.004,0.02]$ we create a foreground map model. We find a remarkable resemblance between this temperature model map based on nearby galaxies and the Planck CMB map. Out of 1000 simulated maps, none of them show such a strong correlation with the foreground map over both large and small angular scales. In particular, the quadrupole, octopole, as well as $\ell=4$ and $\ell=5$ modes correlate with the foreground map to high significance. Furthermore, one of the most prominent temperature decrements in the foreground map coincides with the position of the CMB cold spot. The largest scales of the CMB and thereby the cosmological parameters, may have important changes after proper corrections of this foreground component. However, a reliable corrected CMB map can only be derived when suitable physical mechanisms are proposed and tested.

Bill-of-materials and telecommunications billing applications, need to process both short transactions and long read-write transactions simultaneously. Recent work rarely addresses such evolving workloads. To deal with these workloads, we propose a new concurrency control protocol, Shirakami. Shirakami is a hybrid protocol. The first protocol, Shirakami-LTX, is for long read-write transactions based on multiversion view serializability. The second protocol, Shirakami-OCC, is for short transactions based on Silo. Shirakami naturally integrates them with the write-preservation and epoch-based synchronization. It does not require dynamic protocol switching and provides stable performance. We implemented Shirakami as the transaction processing module of the Tsurugi system, which is a production-grade relational database system. The experimental results demonstrated that Tsurugi exhibited 19.7 times lower latency than PostgreSQL, and Shirakami-LTX exhibited 680 times higher throughput than Shirakami-OCC.

We develop the theory of quantum (a.k.a. noncommutative) relations and quantum (a.k.a. noncommutative) graphs in the finite-dimensional covariant setting, where all systems (finite-dimensional $C^*$-algebras) carry an action of a compact quantum group $G$, and all channels (completely positive maps preserving the canonical $G$-invariant state) are covariant with respect to the $G$-actions. We motivate our definitions by applications to zero-error quantum communication theory with a symmetry constraint. Some key results are the following: 1) We give a necessary and sufficient condition for a covariant quantum relation to be the underlying relation of a covariant channel. 2) We show that every quantum confusability graph with a $G$-action (which we call a quantum $G$-graph) arises as the confusability graph of a covariant channel. 3) We show that a covariant channel is reversible precisely when its confusability $G$-graph is discrete. 4) When $G$ is quasitriangular (this includes all compact groups), we show that covariant zero-error source-channel coding schemes are classified by covariant homomorphisms between confusability $G$-graphs.

Interrupted time series (ITS) is often used to evaluate the effectiveness of a health policy intervention that accounts for the temporal dependence of outcomes. When the outcome of interest is a percentage or percentile, the data can be highly skewed, bounded in $[0, 1]$, and have many zeros or ones. A three-part Beta regression model is commonly used to separate zeros, ones, and positive values explicitly by three submodels. However, incorporating temporal dependence into the three-part Beta regression model is challenging. In this article, we propose a marginalized zero-one-inflated Beta time series model that captures the temporal dependence of outcomes through copula and allows investigators to examine covariate effects on the marginal mean. We investigate its practical performance using simulation studies and apply the model to a real ITS study.

Optimal paths for the classical Onsager-Machlup function determining most probable paths between points on a manifold are only explicitly identified for specific processes, for example the Riemannian Brownian motion. This leaves out large classes of manifold-valued processes such as processes with parallel transported non-trivial diffusion matrix, processes with rank-deficient generator and sub-Riemannian processes, and push-forwards to quotient spaces. In this paper, we construct a general approach to definition and identification of most probable paths by measuring the Onsager-Machlup function on the anti-development of such processes. The construction encompasses large classes of manifold-valued process and results in explicit equation systems for the paths that we denote \emph{development most probable paths}. We define and derive these results and apply them to several cases of stochastic processes on Lie groups, homogeneous spaces, and landmark spaces appearing in shape analysis.

In this work, the cross derivative of the Gibbs free energy, initially proposed for phase transitions in classical spin models [Phys. Rev. B 101, 165123 (2020)], is extended for quantum systems. We take the spin-1 XXZ chain with anisotropies as an example to demonstrate its effectiveness and convenience for the Gaussian-type quantum phase transitions therein. These higher-order transitions are very challenging to determine by conventional methods. From the cross derivative with respect to the two anisotropic strengths, a single valley structure is observed clearly in each system size. The finite-size extrapolation of the valley depth shows a perfect logarithmic divergence, signaling the onset of a phase transition. Meanwhile, the critical point and the critical exponent for the correlation length are obtained by a power-law fitting of the valley location in each size. The results are well consistent with the best estimations in the literature. Its application to other quantum systems with continuous phase transitions is also discussed briefly.

Any knot diagram can be transformed into the unknot by a series of unknotting operations. This paper introduces the diagonal move, a novel unknotting operation that generalizes and unifies several existing moves. We prove that the diagonal move is an efficient unknotting operation for both classical and welded knots, demonstrating that any knot or link can be reduced to the unknot or unlink via a finite sequence of diagonal moves and Reidemeister moves. Additionally, we analyze the distance between knots under diagonal moves, showing that it often requires fewer operations than traditional crossing changes, and extend our results to welded knots, confirming the diagonal move's applicability in this broader setting. Our findings provide a powerful new tool for knot simplification and equivalence, advancing topological and combinatorial knot theory.

We report and analyse the presence of foregrounds in the cosmic microwave background (CMB) radiation associated to extended galactic halos. Using the cross correlation of Planck and WMAP maps and the 2MRS galaxy catalogue, we find that the mean temperature radial profiles around nearby galaxies at $cz\le 4500~\rm{km~s^{-1}}$ show a statistically significant systematic decrease of $\sim 15~μ\rm{K}$ extending up to several galaxy radii. This effect strongly depends on the galaxy morphological type at scales within several tens of times the galaxy size, becoming nearly independent of galaxy morphology at larger scales. The effect is significantly stronger for the more extended galaxies, with galaxy clustering having a large impact on the results. Our findings indicate the presence of statistically relevant foregrounds in the CMB maps that should be considered in detailed cosmological studies. Besides, we argue that these can be used to explore the intergalactic medium surrounding bright late-type galaxies and allow for diverse astrophysical analyses.

Clinical and epidemiological studies encode participant information in multivariate vectors with mixed type variables on continuous, truncated, ordinal, and binary scales. Semiparametric Gaussian Copula (SGC) assumes that observed data is generated by latent multivariate normal random variables which marginals are monotonically transformed and then truncated/ordinalized/binarized. In SGC, the latent correlation matrix fully determines the dependence structure and it is estimated through an inversion of ``bridges'' between Kendall's Tau rank correlations of observed variables and latent correlations. By employing SGC, we develop regression (SGC-Reg), principal component analysis (SGC-PCA), and principal component regression (SGC-PCR) for latent representations of observed data. To build our framework, we make several key contributions: i) establishing novel bridging results for general ordinal type variables, ii) developing regression estimation on the latent space and deriving asymptotic normality of estimators, iii) developing a computationally efficient algorithm that reduces calculation complexity of all steps including calculation of asymptotic covariance matrix from $O(n^4)$ to $O(n\log n)$, iv) developing methods to predict latent representations of observed data and perform imputation of missing data, and v) developing principal component analysis and principal component regression on the latent space. We apply our framework to study the association between a 5-year mortality and 61 frailty-related measures composed of 29 continuous, 17 ordinal, and 15 binary variables in 9478 participants of 1999-2010 waves of National Health and Nutrition Examination Survey (NHANES).

The symplectic spectral metric on the set of Lagrangian submanifolds or Hamiltonian maps can be used to define a completion of these spaces. For an element of such a completion, we define its $γ$-support. We also define the notion of $γ$-coisotropic set, and prove that a $γ$-support must be $γ$-coisotropic toghether with many properties of the $γ$-support and $γ$-coisotropic sets. We give examples of Lagrangians in the completion having large $γ$-support and we study those (called "regular Lagrangians") having small $γ$-support. We compare the notion of $γ$-coisotropy with other notions of isotropy.

Spontaneous structural rearrangements play a central role in the organization and function of complex biomolecular systems. In principle, physics-based computer simulations like Molecular Dynamics (MD) enable us to investigate these thermally activated processes with an atomic level of resolution. However, rare conformational transitions are intrinsically hard to investigate with MD, because an exponentially large fraction of computational resources must be invested to simulate thermal fluctuations in metastable states. Path sampling methods like Transition Path Sampling hold the great promise of focusing the available computational power on sampling the rare stochastic transition between metastable states. In these approaches, one of the outstanding limitations is to generate paths that visit significantly different regions of the conformational space at a low computational cost. To overcome these problems we introduce a rigorous approach that integrates a machine learning algorithm and MD simulations implemented on a classical computer with adiabatic quantum computing. First, using functional integral methods, we derive a rigorous low-resolution representation of the system's dynamics, based on a small set of molecular configurations generated with machine learning. Then, a quantum annealing machine is employed to explore the transition path ensemble of this low-resolution theory, without introducing un-physical biasing forces to steer the system's dynamics. Using the D-Wave quantum computer, we validate our scheme by simulating a benchmark conformational transition in a state-of-the-art atomistic description. We show that the quantum computing step generates uncorrelated trajectories, thus facilitating the sampling of the transition region in configuration space. Our results provide a new paradigm for MD simulations to integrate machine learning and quantum computing.

We design a fast algorithm that computes, for a given linear differential operator with coefficients in $Z[x ]$, all the characteristic polynomials of its p-curvatures, for all primes $p < N$ , in asymptotically quasi-linear bit complexity in N. We discuss implementations and applications of our algorithm. We shall see in particular that the good performances of our algorithm are quickly visible.

One construction of the Alexander polynomial is as a quantum invariant associated with representations of restricted quantum $\mathfrak{sl}_2$ at a fourth root of unity. We generalize this construction to define a link invariant $Δ_{\mathfrak{g}}$ for any semisimple Lie algebra $\mathfrak{g}$ of rank $n$, taking values in $n$-variable Laurent polynomials. Focusing on the case $\mathfrak{g}=\mathfrak{sl}_3$, we establish a direct relation between $Δ_{\mathfrak{sl}_3}$ and the Alexander polynomial. We show that certain parameter evaluations of $Δ_{\mathfrak{sl}_3}$ recover the Alexander polynomial on knots, despite the $R$-matrix not satisfying the Alexander-Conway skein relation at these points. We tabulate $Δ_{\mathfrak{sl}_3}$ for all knots up to seven crossings and various other examples, including the Kinoshita-Terasaka knot and Conway knot mutant pair which are distinguished by this invariant.

A Lie (super)algebra with a non-degenerate invariant symmetric bilinear form $B$ is called a nis-(super)algebra. The double extension $\mathfrak{g}$ of a nis-(super)algebra $\mathfrak{a}$ is the result of simultaneous adding to $\mathfrak{a}$ a central element and a derivation so that $\mathfrak{g}$ is a nis-algebra. Loop algebras with values in simple complex Lie algebras are most known among the Lie (super)algebras suitable to be doubly extended. In characteristic 2 the notion of double extension acquires specific features. Restricted Lie (super)algebras are among the most interesting modular Lie superalgebras. In characteristic 2, using Grozman's Mathematica-based package SuperLie, we list double extensions of restricted Lie superalgebras preserving the non-degenerate closed 2-forms with constant coefficients. The results are proved for the number of indeterminates ranging from 4 to 7 - sufficient to conjecture the pattern for larger numbers. Considering multigradings allowed us to accelerate computations up to 100 times.

We study the properties of the analogue of R-diagonal operators in the setting of bi-free probability. Products of bi-R-diagonal pairs of operators that are $*$-bi-free are studied and powers of such pairs are found to also be bi-R-diagonal. It is moreover shown that the joint $*$-distribution of a bi-R-diagonal pair of operators remains invariant under the multiplication by a $*$-bi-free bi-Haar unitary pair and equivalent characterizations of bi-R-diagonal pairs are developed.

A measurable relation algebra is a relation algebra in which the identity element is a sum of atoms that can be measured in the sense that the "size" of each such atom can be defined in an intuitive and reasonable way (within the framework of the first-order theory of relation algebras). A large class of examples of such algebras, using systems of groups and coordinated systems of isomorphisms between quotients of the groups, has been constructed. This class of group relation algebras is not large enough to exhaust the class of all measurable relation algebras. In the present article, the class of examples of measurable relation algebras is considerably extended by adding one more ingredient to the mix: systems of cosets that are used to "shift" the operation of relative multiplication. It is shown that, under certain additional hypotheses on the system of cosets, each such coset relation algebra with a shifted operation of relative multiplication is an example of a measurable relation algebra. We also show that the class of coset relation algebras does contain examples that are not representable as set relation algebras. In a later article, it is shown that the class of coset relation algebras is adequate to the task of describing all measurable relation algebras in the sense that every atomic measurable relation algebra is essentially isomorphic to a coset relation algebra.

We establish a duality between monads and monadic morphisms in any $(\infty,2)$-category and characterize monadic morphisms in a wide class of examples. This duality unifies several dualities between algebraic structures and their representations, and provides a general mechanism for transferring structure from a monad to its $\infty$-category of algebras. This transfer of structure yields uniform constructions of tensor products for algebras over lax symmetric monoidal and oplax symmetric monoidal monads, extending classical tensor products for modules and operadic algebras. Using this framework, we construct a relative tensor product for algebras over lax monoidal monads, a tensor product for algebras over Hopf $\infty$-operads and equip the $\infty$-category of operadic algebras with canonical enrichment.

Albertson and Berman conjectured that every planar graph has an induced forest on half of its vertices. The best known lower bound, due to Borodin, is that every planar graph has an induced forest on two fifths of its vertices. In a related result, Chartran and Kronk, proved that the vertices of every planar graph can be partitioned into three sets, each of which induce a forest. We show tighter results for 2-outerplanar graphs. We show that every 2-outerplanar graph has an induced forest on at least half the vertices by showing that its vertices can be partitioned into two sets, each of which induces a forest. We also show that every 2-outerplanar graph has an induced outerplanar graph on at least two-thirds of its vertices, assuming that the connected components of the inner layer are two-connected.

We prove an equivalence result between the validity of a pointwise Hardy inequality in a domain and uniform capacity density of the complement. This result is new even in Euclidean spaces, but our methods apply in general metric spaces as well. We also present a new transparent proof for the fact that uniform capacity density implies the classical integral version of the Hardy inequality in the setting of metric spaces. In addition, we consider the relations between the above concepts and certain Hausdorff content conditions.