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Macaulay's theorem and Fr\"oberg's conjecture deal with the Hilbert function of homogeneous ideals in polynomial rings $S$ over a field $K$. In this short note we present some questions related to variants of Macaulay's theorem and Fr\"oberg's conjecture for $K$-subalgebras of polynomial rings. In details, given a subspace $V$ of forms of degree $d$ we consider the $K$-subalgebra $K[V]$ of $S$ generated by $V$. What can be said about Hilbert function of $K[V]$? The analogy with the ideal case suggests several questions. To state them we start by recalling Macaulay's theorem, Fr\"oberg's conjecture and Gotzmann's persistence theorem for ideals. Then we presents the variants for $K$-subalgebras along with some partial results and examples., Comment: A short note written to celebrate the 50-th anniversary of the "Rendiconti dell'Istituto di Matematica dell'Universit\`a di Trieste". To appear in Rend. Istit. Mat. Univ. Trieste Volume 50 (2018)

Linear regression without correspondences is the problem of performing a linear regression fit to a dataset for which the correspondences between the independent samples and the observations are unknown. Such a problem naturally arises in diverse domains such as computer vision, data mining, communications and biology. In its simplest form, it is tantamount to solving a linear system of equations, for which the entries of the right hand side vector have been permuted. This type of data corruption renders the linear regression task considerably harder, even in the absence of other corruptions, such as noise, outliers or missing entries. Existing methods are either applicable only to noiseless data or they are very sensitive to initialization or they work only for partially shuffled data. In this paper we address these issues via an algebraic geometric approach, which uses symmetric polynomials to extract permutation-invariant constraints that the parameters $\xi^* \in \Re^n$ of the linear regression model must satisfy. This naturally leads to a polynomial system of $n$ equations in $n$ unknowns, which contains $\xi^*$ in its root locus. Using the machinery of algebraic geometry we prove that as long as the independent samples are generic, this polynomial system is always consistent with at most $n!$ complex roots, regardless of any type of corruption inflicted on the observations. The algorithmic implication of this fact is that one can always solve this polynomial system and use its most suitable root as initialization to the Expectation Maximization algorithm. To the best of our knowledge, the resulting method is the first working solution for small values of $n$ able to handle thousands of fully shuffled noisy observations in milliseconds., Comment: 13 pages

We present angle dependent measurements of the damping properties of epitaxial Fe layers with MgO, Al and Pt capping layers. Based on the preferential distribution of lattice defects following the crystal symmetry, we make use of a model of the defect density to separate the contribution of two-magnon scattering to the damping from the isotropic contribution originating in the spin pumping effect, the viscous Gilbert damping and the magnetic proximity effect. The separation of the two-magnon contribution, which depends strongly on the defect density, allows for the measurement of a value of the effective spin mixing conductance which is closer to the value exclusively due to spin pumping. The influence of the defect density for bilayers systems due to the different capping layers and to the unavoidable spread in defect density from sample to sample is thus removed. This shows the potential of studying spin pumping phenomena in fully ordered systems in which this separation is possible, contrary to polycrystalline or amorphous metallic thin films., Comment: 7 pages, 3 figures

Let I be a homogeneous ideal of a polynomial ring S. We prove that if the initial ideal J of I, w.r.t. a term order on S, is square-free, then the extremal Betti numbers of S/I and of S/J coincide. In particular, depth(S/I)=depth(S/J) and reg(S/I)=reg(S/J)., Comment: Minor changes throughout. A compressed version of the paper will appear in Inventions Mathematicae

We consider the Robin boundary value problem $\mathrm{div} (A \nabla u) = \mathrm{div} \mathbf{f}+F$ in $\Omega$, $\mathcal{C}^1$ domain, with $(A \nabla u - \mathbf{f})\cdot \mathbf{n} + \alpha u = g$ on $\Gamma$, where the matrix $A$ belongs to $VMO (\mathbb{R}^3) $, and discover the uniform estimates on $\|u\|_{W^{1,p}(\Omega)}$, with $1 < p < \infty$, independent on $\alpha$. At the difference with the case $p = 2,$ which is simpler, we call here the weak reverse H\"older inequality. This estimates show that the solution of Robin problem converges strongly to the solution of Dirichlet (resp. Neumann) problem in corresponding spaces when the parameter $\alpha$ tends to $\infty$ (resp. $0$)., Comment: corrected proof of Theorem 2.3

We study the stationary Stokes and Navier-Stokes equations with non-homogeneous Navier boundary condition in a bounded domain $\Omega\subset\mathbb{R}^{3}$ of class $\mathcal{C}^{1,1}$. We prove existence, uniqueness of weak and strong solutions in $\mathbf{W}^{1,p}(\Omega)$ and $\mathbf{W}^{2,p}(\Omega)$ for all $1

Motivated by questions in algebra and combinatorics we study two ideals associated to a simple graph G: --> the Lovasz-Saks-Schrijver ideal defining the d-dimensional orthogonal representations of the graph complementary to G and --> the determinantal ideal of the (d+1)-minors of a generic symmetric with 0s in positions prescribed by the graph G. In characteristic 0 these two ideals turns out to be closely related and algebraic properties such as being radical, prime or a complete intersection transfer from the Lovasz-Saks-Schrijver ideal to the determinantal ideal. For Lovasz-Saks-Schrijver ideals we link these properties to combinatorial properties of G and show that they always hold for d large enough. For specific classes of graph, such a forests, we can give a complete picture and classify the radical, prime and complete intersection Lovasz-Saks-Schrijver ideals., Comment: Paper restructured

We report on the determination of the spin Hall angle and the identification of the role of the interface roughness for the inverse spin Hall effect (ISHE) in ultra-clean, defect-reduced epitaxial Pt films. By applying vector network analyzer ferromagnetic resonance spectroscopy to a series of single crystalline Fe (12 nm) /Pt (t$_\text{Pt}$) bilayers we determine the real part of the spin mixing conductance $(4.4\:\pm\:0.2)\cdot 10^{19}\: $m$^{-2}$ and reveal a very small spin diffusion length in the epitaxial Pt $(1.1\:\pm\:0.1)$ nm film. We investigate the spin pumping and ISHE in a stripe microstucture excited by a microwave coplanar waveguide (CPW) antenna. By using their different angular dependencies, we distinguish between spin rectification effects and the inverse spin Hall effect. The relatively large value of the spin Hall angle $(5.7 \pm 1.4)\: \% $ hints that interface engineering by epitaxially grown non-magnetic materials can increase the spin-to-charge current conversion efficiency.

We show that the Koszul homology algebra of the second Veronese subalgebra of a polynomial ring over a field of characteristic zero is generated, as an algebra, by the homology classes corresponding to the syzygies of the lowest linear strand., Comment: 7 pages

We present algebraic and geometric arguments that give a complete classification of the rational normal scrolls that are hyperplane section of a given rational normal scrolls., Comment: To appear in the special issue of the Bull. Math. Soc. Sci. Math. Roumanie honoring Prof. Dorin Popescu

Hankel determinantal rings, i.e., determinantal rings defined by minors of Hankel matrices of indeterminates, arise as homogeneous coordinate rings of higher order secant varieties of rational normal curves; they may also be viewed as linear specializations of generic determinantal rings. We prove that, over fields of characteristic zero, Hankel determinantal rings have rational singularities; in the case of positive prime characteristic, we prove that they are F-pure. Independent of the characteristic, we give a complete description of the divisor class groups of these rings, and show that each divisor class group element is the class of a maximal Cohen-Macaulay module.

We investigate thickness series of films of the Heusler alloy Co$_2$FeAl in order to study the effect of annealing on the interface with a MgO layer and on the bulk magnetic properties. Our results reveal that while the perpendicular interface anisotropy constant $K^{\perp}_{\rm S}$ is zero for the as-deposited samples, its value increases with annealing up to a value of $1.14\, \pm \,0.07$~mJ/m$^2$ for the series annealed at 320$^{\rm o}$C and of $2.07\, \pm \,0.7$~mJ/m$^2$ for the 450$^{\rm o}$C annealed series owing to a strong modification of the interface during the thermal treatment. This large value ensures a stabilization of a perpendicular magnetization orientation for a thickness below 1.7~nm. The data additionally shows that the in-plane biaxial anisotropy constant has a different evolution with thickness in as-deposited and annealed systems. The Gilbert damping parameter $\alpha$ shows minima for all series for a thickness of 40~nm and an absolute minimum value of $2.8\pm0.1\cdot10^{-3}$. The thickness dependence is explained in terms of an inhomogenous magnetization state generated by the interplay between the different anisotropies of the system and by crystalline disorder., Comment: 6 pages, 5 figures

Conca and Herzog proved that any product of ideals of linear forms in a polynomial ring has a linear resolution. The goal of this paper is to establish the same result for any quadric hypersurface. The main tool we develop and use is a flexible version of Derksen and Sidman's approximation systems., Comment: 14 pages

This work concerns the moment map $\mu$ associated with the standard representation of a classical Lie algebra. For applications to deformation quantization it is desirable that $S/(\mu)$, the coordinate algebra of the zero fibre of $\mu$, be Koszul. The main result is that this algebra is not Koszul for the standard representation of $\mathfrak{sl}_{n}$, and of $\mathfrak{sp}_{n}$. This is deduced from a computation of the Betti numbers of $S/(\mu)$ as an $S$-module, which are of interest also from the point of view of commutative algebra., Comment: Revised version. Differences to version 1: title slightly changed, comments added at the end, minor revisions

Inspired by work of Cartwright and Sturmfels, in a previous paper we introduced two classes of multigraded ideals named after them. These ideals are defined in terms of properties of their multigraded generic initial ideals. The goal of this paper is showing that three families of ideals that have recently attracted the attention of researchers are Cartwright-Sturmfels ideals. More specifically, we prove that binomial edge ideals, multigraded homogenizations of linear spaces, and multiview ideals are Cartwright-Sturmfels ideals, hence recovering and extending recent results of Herzog-Hibi-Hreinsdottir-Kahle-Rauh, Ohtani, Ardila-Boocher, Aholt-Sturmfels-Thomas, and Binglin Li. We also propose a conjecture on the rigidity of local cohomology modules of Cartwright-Sturmfels ideals, that was inspired by a theorem of Brion. We provide some evidence for the conjecture by proving it in the monomial case., Comment: 22 pages. We are grateful to Giulia Gaggero for pointing out a mistake in the proof of Theorem 2.1, which we corrected in the current version

We investigate the effect of Al doping on the magnetic properties of the alloy CoFeB. Comparative measurements of the saturation magnetization, the Gilbert damping parameter $\alpha$ and the exchange constant as a function of the annealing temperature for CoFeB and CoFeAlB thin films are presented. Our results reveal a strong reduction of the magnetization for CoFeAlB in comparison to CoFeB. If the prepared CoFeAlB films are amorphous, the damping parameter $\alpha$ is unaffected by the Al doping in comparison to the CoFeB alloy. In contrast, in the case of a crystalline CoFeAlB film, $\alpha$ is found to be reduced. Furthermore, the x-ray characterization and the evolution of the exchange constant with the annealing temperature indicate a similar crystallization process in both alloys. The data proves the suitability of CoFeAlB for spin torque switching properties where a reduction of the switching current in comparison with CoFeB is expected., Comment: 5 Pages, 5 Figures

The spin-pumping mechanism is probed through a tunnelling MgO interlayer in Fe/Pt bilayers. We show by ferromagnetic resonance technique and spin-pumping experiments that spin currents can tunnel through the MgO interlayer for thickness up to 2~{nm} and can produce significant voltages in the Pt layer. The electrical detection of spin-pumping furthermore reveals the critical role of rectification and shunting effects on the generated voltages. The non zero spin current transport through a few monolayers of an insulating interlayer might initiate further studies on the role of very thin oxides in spin-pumping experiments., Comment: 5 pages, 4 figures

We quantify the relative weight of inverse spin Hall and spin rectification effects occurring in RF-sputtered polycrystalline permalloy, molecular beam epitaxy-grown epitaxial iron and liquid phase epitaxy-grown yttrium-iron-garnet bilayer systems with different capping materials. To distinguish the spin rectification signal from the inverse spin Hall voltage the external magnetic field is rotated in-plane to take advantage of the different angular dependencies of the prevailing effects. We prove that in permalloy anisotropic magnetoresistance is the dominant source for spin rectification while in epitaxial iron the anomalous Hall effect has an also comparable strength. The rectification in yttrium-iron-garnet/platinum bilayers reveals an angular dependence imitating the one seen for anisotropic magnetoresistance caused by spin Hall magnetoresistance., Comment: 6 pages, 6 figures

We investigate spin pumping phenomena in polycrystalline CoFeB/Pt and CoFeB/Ta bilayers and the correlation between the effective spin mixing conductance $g^{\uparrow\downarrow}_{\rm eff}$ and the obtained voltages generated by the spin-to-charge current conversion via the inverse spin Hall effect in the Pt and Ta layers. For this purpose we measure the in-plane angular dependence of the generated voltages on the external static magnetic field and we apply a model to separate the spin pumping signal from the one generated by the spin rectification effect in the magnetic layer. Our results reveal a dominating role of anomalous Hall effect for the spin rectification effect with CoFeB and a lack of correlation between $g^{\uparrow\downarrow}_{\rm eff}$ and inverse spin Hall voltages pointing to a strong role of the magnetic proximity effect in Pt in understanding the observed increased damping. This is additionally reflected on the presence of a linear dependency of the Gilbert damping parameter on the Pt thickness., Comment: 6 Pages, 5 Figures

New title and minor adjustments. To appear in the Journal of Pure and Applied Algebra, Comment: Dedicated to the memory of our friend Tony Geramita

Let I be either the ideal of maximal minors or the ideal of 2-minors of a row graded or column graded matrix of linear forms L. In two previous papers we showed that I is a Cartwright-Sturmfels ideal, that is, the multigraded generic initial ideal gin(I) of I is radical (and essentially independent of the term order chosen). In this paper we describe generators and prime decomposition of gin(I) in terms of data related to the linear dependences among the row or columns of the submatrices of L. In the case of 2-minors we also give a closed formula for its multigraded Hilbert series., Comment: 14 pages

We describe the universal Groebner basis of the ideal of maximal minors and the ideal of $2$-minors of a multigraded matrix of linear forms. Our results imply that the ideals are radical and provide bounds on the regularity. In particular, the ideals of maximal minors have linear resolutions. Our main theoretical contribution consists of introducing two new classes of ideals named after Cartwright and Sturmfels, and proving that they are closed under multigraded hyperplane sections. The gins of the ideals that we study enjoy special properties., Comment: 10 pages

In this paper, we introduce the macroscopic quantity, namely the dispersion tensor or the \textit{Burnett coefficient}s in the class of generalized Hashin-Shtrikman micro-structures \cite[page no. 281]{T}. In the case of two-phase materials associated with the periodic Hashin-Shtrikman structures, we settle the issue that the dispersion tensor has an unique minimizer, which is so called Apollonian-Hashin-Shtrikman micro-structure.

In this paper, we use spectral methods by introducing the Bloch waves to study the homogenization process in the non-periodic class of generalized Hashin-Shtrikman micro-structures \cite[page no. 281]{T}, which incorporates both translation and dilation with a family of scales, including one subclass of laminates. We establish the classical homogenization result with providing the spectral representation of the homogenized coefficients. It offers a new lead towards extending the Bloch spectral analysis in the non-periodic, non-commutative class of micro-structures.

The goal of this paper is to present examples of families of homogeneous ideals in the polynomial ring over a field that satisfy the following condition: every product of ideals of the family has a linear free resolution. As we will see, this condition is strongly correlated to good primary decompositions of the products and good homological and arithmetical properties of the associated multi-Rees algebras. The following families will be discussed in detail: polymatroidal ideals, ideals generated by linear forms and Borel fixed ideals of maximal minors. The main tools are Gr\"obner bases and Sagbi deformation.

We study a large family of products of Borel fixed ideals of maximal minors. We compute their initial ideals and primary decompositions, and show that they have linear free resolutions. The main tools are an extension of straightening law and a very surprising primary decomposition formula. We study also the homological properties of associated multi-Rees algebra which are shown to be Cohen-Macaulay, Koszul and defined by a Gr\"obner basis of quadrics.

We study absolutely Koszul algebras, Koszul algebras with the Backelin-Roos property and their behavior under standard algebraic operations. In particular, we identify some Veronese subrings of polynomial rings that have the Backelin-Roos property and conjecture that the list is indeed complete. Among other things, we prove that every universally Koszul ring defined by monomials has the Backelin-Roos property.

Inspired by recent results of Ein, Lazarsfeld, Erman and Zhou on the non-vanishing of Betti numbers of high Veronese subrings, we describe the behaviour of the Betti numbers of Stanley-Reisner rings associated with iterated barycentric or edgewise subdivisions of a given simplicial complex. Our results show that for a simplicial complex $\Delta$ of dimension $d-1$ and for $1\leq j\leq d-1$ the number of $0$'s the j-th linear strand of the minimal free resolution of the r-th barycentric or edgewise subdivision is bounded above only in terms of $d$ and $j$ (and independently of $r$)., Comment: changed title, 32 pages

We show that the ideal generated by the $(n-2)$ minors of a general symmetric $n$ by $n$ matrix has an initial ideal that is the Stanley-Reisner ideal of the boundary complex of a simplicial polytope and has the same Betti numbers.

Measurements of the exchange stiffness $D$ and the exchange constant $A$ of Yttrium Iron Garnet (YIG) films are presented. The YIG films with thicknesses from 0.9 $\mu$m to 2.6 $\mu$m were investigated with a microwave setup in a wide frequency range from 5 to 40 GHz. The measurements were performed when the external static magnetic field was applied in-plane and out-of-plane. The method of Schreiber and Frait, based on the analysis of the perpendicular standing spin wave (PSSW) mode frequency dependence on the applied out-of-plane magnetic field, was used to obtain the exchange stiffness $D$. This method was modified to avoid the influence of internal magnetic fields during the determination of the exchange stiffness. Furthermore, the method was adapted for in-plane measurements as well. The results obtained using all methods are compared and values of $D$ between $(5.18\pm0.01) \cdot 10^{-17}$T$\cdot$m$^2$ and $(5.34\pm0.02) \cdot 10^{-17}$ T$\cdot$m$^2$ were obtained for different thicknesses. From this the exchange constant was calculated to be $A=(3.65 \pm 0.38)~$pJ/m., Comment: 10 pages, 3 figures, 3 tables

We consider the problem of distributing two conducting materials with a prescribed volume ratio in a given domain so as to minimize the first eigenvalue of an elliptic operator with Dirichlet conditions. The gap between the two conductivities is assumed to be small (low contrast regime). For any geometrical configuration of the mixture, we provide a complete asymptotic expansion of the first eigenvalue. We then consider a relaxation approach to minimize the second order approximation with respect to the mixture. We present numerical simulations in dimensions two and three.