Ideal Resolution Research Articles

Resolutions of ideals of subspace arrangements

Given a collection of t subspaces in an n-dimensional vector space W we can associate to them t linear ideals in the symmetric algebra 𝒮(W∗). Conca and Herzog showed that the Castelnuovo–Mumford regularity of the product of t linear ideals is equal to t. Derksen and Sidman showed that the Castelnuovo–Mumford regularity of the intersection of t linear ideals is at most t. We show that analogous results hold when we work over the exterior algebra ⋀(W∗) (over a field of characteristic 0). To prove these results we rely on the functoriality of equivariant free resolutions and construct a functor Ω from the category of polynomial functors to itself. The functor Ω transforms resolutions of polynomial functors associated to subspace arrangements over the symmetric algebra to resolutions over the exterior algebra.

Multispectral and Hyperspectral Image Fusion Based on Regularized Coupled Non-Negative Block-Term Tensor Decomposition

The problem of multispectral and hyperspectral image fusion (MHF) is to reconstruct images by fusing the spatial information of multispectral images and the spectral information of hyperspectral images. Focusing on the problem that the hyperspectral canonical polyadic decomposition model and the Tucker model cannot introduce the physical interpretation of the latent factors into the framework, it is difficult to use the known properties and abundance of endmembers to generate high-quality fusion images. This paper proposes a new fusion algorithm. In this paper, a coupled non-negative block-term tensor model is used to estimate the ideal high spatial resolution hyperspectral images, its sparsity is characterized by adding 1-norm, and total variation (TV) is introduced to describe piecewise smoothness. Secondly, the different operators in two directions are defined and introduced to characterize their piecewise smoothness. Finally, the proximal alternating optimization (PAO) algorithm and the alternating multiplier method (ADMM) are used to iteratively solve the model. Experiments on two standard datasets and two local datasets show that the performance of this method is better than the state-of-the-art methods.

Enantioselective bio-deacylation of arylalkyl acetates using tertiary amines as additive under promiscuous conditions

Herein, we describe the impact of the introduction of tertiary amines as additive during the enzymatic kinetic resolution via deacylation of some arylalkyl acetates 1a-8a under promiscuous conditions. Two CAL-B preparations were examined: Novozym®435 and CHIRAZYME® L-2, c.-f. C2, lyo. The influence of the introduction of seven amines is checked: Triethylamine (Et3N), Pyridine, 4-Dimethylaminopyridine (4-DMAP), 1,4-Diazabicyclo[2.2.2]octane (DABCO), Cinchonine, Cinchonidine, Quinine and Quinidine; and that in two organic solvent: diisopropylether (DIPE) and tertiobutylmethyl ether (TBME).Among the examined amines, the use of DABCO as additive recorded the best results in terms of reactivity and selectivity. It was to be highlighted that this additive was reported as an enzyme activator for the first time in the enzymatic kinetic resolution via hydrolysis of racemic acetates under non-aqueous conditions. Both CAL-B preparations showed the same behavior in the presence of the chosen additives during the biodeacylation of the resolved acetates. No direct correlation between the pKa values of several used additives and the CAL-B activation rates has been revealed.An ideal enzymatic kinetic resolution was recorded with the acetate 4a (Conv 50% and E > >200).

Tunable hyperbolic Cohen-class kernel for cross-term diminishing in time–frequency distributions

Nonstationary signals are found in many engineering applications. The analysis of nonstationary signals trough bi-dimensional functions allows tracing their time and frequency evolution. In this regard, energy-distribution representations allocate the analyzed-signal power over two description variables, time and frequency, and offer a large number of mathematical properties, such as an ideal infinite resolution in both domains. Unfortunately, energy-distribution representations generate cross terms for multi-component nonstationary signals, which might mislead the analysis interpretation. Cross-term effects can be lessened by applying an appropriate smoothing kernel; they distort the shape of auto terms at the same time. In this work, a novel tunable signal-independent hyperbolic kernel function is introduced for reducing cross-term effects, preserving valuable properties, during the analysis of multi-component nonstationary signals. Obtained results from computer-model and real-life cases of study validate and demonstrate the performance superiority of the proposed kernel function, in terms of cross-term suppression and time–frequency resolution, against the well-known and widely used the Choi-Williams distribution and the cone-shape distribution.

Simulation of a micron resolution capillary liquid scintillation detector for 14 MeV fusion neutrons

Aiming to improve the spatial resolution of a neutron imaging system (NIS) for 14 MeV fusion neutrons, an ideal micron resolution capillary detector filled with a high optical index liquid scintillator was simulated. A threshold for each capillary pixel and a threshold for each cluster were applied to suppress the gamma-induced background. In addition, by using a pattern recognition algorithm and an optimized Hough transform, the accuracy of determining the neutron impinging positions and the dynamic range of this detector were enhanced. For an ideal capillary array detector, the spatial resolution is expected as one capillary size of 20μm. The dynamic range of ∼1000 is reachable while the accuracy of neutron impinging position determination keeps better than 85%. The ionization quenching, light sharing and energy resolution of the detector were applied to the simulated data to understand the capillary array detector.

Resolution of ideals associated to subspace arrangements

Let $I_1,\dots,I_n$ be ideals generated by linear forms in a polynomial ring over an infinite field and let $J = I_1 \cdots I_n$. We describe a minimal free resolution of $J$ and show that it is supported on a polymatroid obtained from the underlying representable polymatroid by means of the so-called Dilworth truncation. Formulas for the projective dimension and Betti numbers are given in terms of the polymatroid as well as a characterization of the associated primes. Along the way we show that $J$ has linear quotients. In fact, we do this for a large class of ideals $J_P$, where $P$ is a certain poset ideal associated to the underlying subspace arrangement.

Elementary construction of minimal free resolutions of the Specht ideals of shapes (n − 2,2) and (d,d,1)

For a partition [Formula: see text] of [Formula: see text], let [Formula: see text] be the ideal of [Formula: see text] generated by all Specht polynomials of shape [Formula: see text]. We assume that [Formula: see text]. Then [Formula: see text] is Gorenstein, and [Formula: see text] is a Cohen–Macaulay ring with a linear free resolution. In this paper, we construct minimal free resolutions of these rings. Zamaere et al. [Jack polynomials as fractional quantum Hall states and the Betti numbers of the [Formula: see text]-equals ideal, Commun. Math. Phys. 330 (2014) 415–434] already studied minimal free resolutions of [Formula: see text], which are also Cohen–Macaulay, using highly advanced technique of the representation theory. However, we only use the basic theory of Specht modules, and explicitly describe the differential maps.

Free Resolutions and Generalized Hamming Weights of Binary Linear Codes

In this work, we explore the relationship between the graded free resolution of some monomial ideals and the Generalized Hamming Weights (GHWs) of binary codes. More precisely, we look for a structure that is smaller than the set of codewords of minimal support that provides us with some information about the GHWs. We prove that the first and second generalized Hamming weights of a binary linear code can be computed (by means of a graded free resolution) from a set of monomials associated with a binomial ideal related with the code. Moreover, the remaining weights are bounded above by the degrees of the syzygies in the resolution.

Computing generalized hamming weights of binary linear codes via free resolutions

In this work, we explore the relationship between free resolution of some monomial ideals and Generalized Hamming Weights (GHWs) of binary codes. More precisely, we look for a structure smaller than the set of codewords of minimal support that provides us some information about the GHWs. We prove that the first and second generalized Hamming weight of a binary linear code can be computed (by means of a graded free resolution) from a set of monomials associated to a binomial ideal related with the code. Moreover, the remaining weights are bounded by the Betti numbers for that set.

Linearity of free resolutions of monomial ideals

We study monomial ideals with linear presentation or partially linear resolution. We give combinatorial characterizations of linear presentation for square-free ideals of degree 3, and for primary ideals whose resolutions are linear except for the last step (the “almost linear” case). We also give sharp bounds on Castelnuovo–Mumford regularity and numbers of generators in some cases. It is a basic observation that linearity properties are inherited by the restriction of an ideal to a subset of variables, and we study when the converse holds. We construct fractal examples of almost linear primary ideals with relatively few generators related to the Sierpiński triangle. Our results also lead to classes of highly connected simplicial complexes $$\Delta $$ that cannot be extended to the complete $$\mathrm{dim}\Delta $$ -skeleton of the simplex on the same variables by shelling.

CUSTOMARY APPROACH AND RULE OF LAW BY PANGLIMA LAOT IN RESOLVING FISHERMEN'S DISPUTE IN ACEH

This research aims to examine the effectiveness of Panglima Laot's institution authority in the maritim area to resolve sea disputes and to find an ideal dispute resolution model to be used. This research uses qualitative method by sociological juridical approach in the form of descriptive analytical. The Panglima laot's authority in an effort to assist the government in resolving maritime disputes is a formal authority granted by a law which is attributively regulated in Article 28 of Qanun Aceh Number 10 of 2008 concerning Customary Institutions. However, when dealing with certain cases between fishermen where the regulation is not very clear in the Qanun, Panglima Laot seeks to resolve cases by using traditional approaches, religious values, propriety, a sense of justice and human conscience. Therefore, Panglima Laot needs detailed and written instructions for carrying out tasks in the form of Government Regulations or Qanuns. Including dealing with refugees who come from abroad.

Structure theory for a class of grade 3 homogeneous ideals defining type 2 compressed rings

Let R=k[x,y,z] be a standard graded 3-variable polynomial ring, where k denotes any field. We study grade 3 homogeneous ideals I⊆R defining compressed rings with socle k(−s)⊕k(−2s+1), where s≥3 is some integer. We prove that all such ideals are obtained by a trimming process introduced by Christensen, Veliche, and Weyman (J. Commut. Algebra 11:3 (2019), 325–339). We also construct a general resolution for all such ideals which is minimal in sufficiently generic cases. Using this resolution, we give bounds on the minimal number of generators μ(I) of I depending only on s; moreover, we show these bounds are sharp by constructing ideals attaining the upper and lower bounds for all s≥3. Finally, we study the Tor-algebra structure of R∕I. It is shown that these rings have Tor algebra class G(r) for s≤r≤2s−1. Furthermore, we produce ideals I for all s≥3 and all r with s≤r≤2s−1 such that Soc(R∕I)=k(−s)⊕k(−2s+1) and R∕I has Tor-algebra class G(r), partially answering a question of realizability posed by Avramov (J. Pure Appl. Algebra 216:11 (2012), 2489–2506).

Practical access to (S)-heterocyclic aromatic acetates via CAL-B/Na2CO3-deacylation and Mitsunobu reaction protocol

Herein, we report the preparation of enantiomerically pure forms of 2,3-dihydrobenzofuran-3-ol (1), chroman-4-ol (2), thiochroman-4-ol (3), 1-(furan-2-yl) ethanol (5) and 1-(thiophen-2-yl) ethanol (6), through a kinetic resolution catalysed by Candida antarctica lipase B/Na2CO3 hydrolysis sequence in organic media. The (R)-furnished alcohols and the (S)-remained acetates are recovered enantiopures (ee >99%, E ≫ 200, Conv = 50%). Those ideal enzymatic kinetic resolution (EKRs) are well incorporated to the Mitsunobu inversion protocol in a one pot procedure to give (S)-heterocyclic acetates (1a–3a) in good to high enantiomeric excess (88%–92% ee). Whilst, the (S)-heteroaromatic acetates (5a and 6a) are given with moderate enantiomeric excess (51%–62% ee). All the (S)-acetates are given in good isolated chemical yields (>80%) allowing to overcome the maximum of 50% yield which could be usually reached in a regular kinetic resolution processes.

Evaluating the relationship between data resolution and the accuracy of identified helicopter landing zones (HLZs)

Helicopters provide critical advantages in military operations because of their ability to land at small and unimproved sites. While the military uses models to identify helicopter landing zones (HLZs), little research has been conducted on their accuracy. This study evaluated the performance of an HLZ detection model derived from existing selection criteria that incorporated elevation and land cover data with spatial resolutions ranging from 1 m to 30 m. Multiple HLZs were selected as study sites at three geographically varied locations. The HLZ boundaries identified using the derived model were then compared to surveyed reference boundaries to assess their accuracy. This study found that as the spatial resolution of the data became coarser, accuracy decreased across all sites. However, there were some instances where noticeable increases in error were observed at certain resolutions for some sites. The resolution at which this occurred was always related to the size of features either bounding or located within the landing area. Thus, this study found that the most important consideration when determining ideal resolution for HLZ detection is the geography of the study area. While additional research is needed, this study presents initial findings and a framework upon which future assessments can build.

High-throughput volumetric adaptive optical imaging using compressed time-reversal matrix

Deep-tissue optical imaging suffers from the reduction of resolving power due to tissue-induced optical aberrations and multiple scattering noise. Reflection matrix approaches recording the maps of backscattered waves for all the possible orthogonal input channels have provided formidable solutions for removing severe aberrations and recovering the ideal diffraction-limited spatial resolution without relying on fluorescence labeling and guide stars. However, measuring the full input–output response of the tissue specimen is time-consuming, making the real-time image acquisition difficult. Here, we present the use of a time-reversal matrix, instead of the reflection matrix, for fast high-resolution volumetric imaging of a mouse brain. The time-reversal matrix reduces two-way problem to one-way problem, which effectively relieves the requirement for the coverage of input channels. Using a newly developed aberration correction algorithm designed for the time-reversal matrix, we demonstrated the correction of complex aberrations using as small as 2% of the complete basis while maintaining the image reconstruction fidelity comparable to the fully sampled reflection matrix. Due to nearly 100-fold reduction in the matrix recording time, we could achieve real-time aberration-correction imaging for a field of view of 40 × 40 µm2 (176 × 176 pixels) at a frame rate of 80 Hz. Furthermore, we demonstrated high-throughput volumetric adaptive optical imaging of a mouse brain by recording a volume of 128 × 128 × 125 µm3 (568 × 568 × 125 voxels) in 3.58 s, correcting tissue aberrations at each and every 1 µm depth section, and visualizing myelinated axons with a lateral resolution of 0.45 µm and an axial resolution of 2 µm.

Calibration of Wideband FMCW Polarimetric Radars Operating at Millimeter-Wave Frequencies

This article presents a comprehensive technique for calibrating wideband frequency-modulated continuous-wave (FMCW) fully polarimetric radars employing linear FM (LFM) signal chirps. A detailed system distortion model is developed that accounts for both polarimetric distortions and distortions caused by nonlinearities in the transmitted and received radar chirps over multiple polarization channels. Using the radar responses of two point targets at known distances from the radar, the calibration technique is able to estimate the range-dependent nonlinearities in phase and correct them for all targets at all ranges. This allows the radar to achieve its ideal range resolution. In addition, the calibration technique corrects for channel imbalances in the polarimetric radar using a metallic sphere of known diameter and any depolarizing target. Both numeric simulations and actual polarimetric radar measurements using a wideband FMCW radar operating over 76.5–82 GHz were used to validate the calibration technique.

On-chip all-optical quantizer based on the cross polarization modulation effect in a double-ridge a-Si:H-Si3N4 waveguide.

The on-chip all-optical quantizer is a key functional module to realize all-optical analog-to-digital conversion. It has important applications in future optical communication and ultrafast all-optical signal processing systems. To the best of our knowledge, a novel all-optical quantization scheme based on the cross-polarization modulation effect in a double-ridge a-Si:H-Si3N4 waveguide is proposed. For the optimized waveguide, the maximum nonlinear coefficient and transmission loss at the probe wavelength are 2680.5m-1W-1 and 2.246 dB/cm, respectively. The simulation results of the proposed all-optical quantizer show that a nonlinear phase shift of 8 can be generated in the 8.79 mm long waveguide when the peak pump light power is 0.8 W. A 16-level quantization and a 4-bit quantization resolution are obtained. The effective number of bits is 3.79-bit, which is only 0.21-bit away from the ideal resolution.

Multigraded Betti numbers of Möbius domino ideals

We enumerate graded Betti numbers of the minimal free resolutions of ideals generated by domino tilings of Möbius bands by using combinatorial information from the domino tilings in two ways: first using a splitting function to find a recursive formula and second using a bijection to enumerate in terms of binomial coefficients and Fibonacci numbers.

Minimal free resolutions of certain equigenerated monomial ideals

Let R=k[x1,…,xn] denote the standard graded polynomial ring over a field k. We study certain classes of equigenerated monomial ideals with the property that the so-called complementary ideal has no linear relations on the generators. We then use iterated trimming complexes to deduce Betti numbers for such ideals. Furthermore, using a result on splitting mapping cones by Miller and Rahmati, we construct the minimal free resolutions for all ideals under consideration explicitly and conclude with questions about extra structure on these complexes.

Geographical Analysis of Ruralization Phenomenon in Rania City and Impact on Their Urban Enviroment

Ruralization of Urban areas is one of the most important topics of Urban Geography as it is one of the issues of urban areas and a major contributor to other urban issues; therefore, researchers need to look at this issue with caution. Ranya city has experienced this issue to a great extent mostly due to population growth and urban expansion. Ruralization has caused major issues such as infrastructure issues in terms of education, health, transportation and increased crime rates in different parts of the city.
 This research addresses this issue and aims to investigate the causes and characteristics of ruralization in Ranya city. This effort will help the local government to look for solution to address this issue. This research has used descriptive and analytical methods.The research concludes that ruralization has caused some drastic consequences when it comes to providing and delivering different public and infrastructure services. The paper recommends to authorities that preparing a master plan for the city would be the ideal resolution for this issue and other issues in the city.