Projective Dimension Research Articles

Projective Dimension of Some Graphs

In this paper exact values for the projective dimension of edge ideals associated to some star related graphs and product graphs $G\ \square\ P_2$, when $G=\ C_n,\ K_n$ and upper bounds for the projective dimension when $G=\ P_n,\ W_n$, are obtained. We have proved that $pd(C_{n+1}\ \square\ P_2)= 2\big(n-\left\lfloor \frac{n}{4}\right\rfloor\big)$, $pd(K_n\ \square\ P_2)= 2n-2$ and $pd(P_{n+1}\ \square \ P_2)\le n+3+\left\lfloor \frac{n-3}{2}\right\rfloor$, $pd(W_n\ \square\ P_2)\leq n+1+\lceil\frac{2n-1}{3}\rceil$. These values are functions of the number of vertices in the corresponding graphs.

First–principles study of β′ phase in Mg–RE alloys

Resource conservation and environmental protection have become a priority development trend of rail transit. As magnesium alloy is the most effective lightweight material for its low density, high strength, and strong shock absorption, its large–scale application in rail vehicles contributes to lightweight, energy saving, and emission reduction. It is imperative to vigorously develop magnesium alloy with better properties, which may be closely related to its precipitate phases. However, research on the mechanical properties of β′ phase of different Mg–RE alloys is still insufficient. Therefore, the structure of β′ phase in Mg–RE (RE = Y, Gd, Tb, or Dy) alloys is constructed and its formation energies, electronic structures, elastic moduli, and elastic anisotropy were calculated by the first–principles method in this paper. The calculation results of formation enthalpy and elastic constants verify that these four alloys (Mg–Y, Mg–Gd, Mg–Tb, and Mg–Dy) are thermodynamically and mechanically stable. Moreover, the calculations of elastic modulus demonstrate that with RE addition, the elastic modulus are obviously improved, especially in Young's modulus which improvs from 9 to 18 GPa. Vickers hardness of β′ phase in Mg–RE alloys is calculated by the elastic moduli, which shows that Mg–RE alloys exhibit better hardness than that of α–Mg. The elastic anisotropy for each Mg–RE alloy is characterized by elastic anisotropic factors, three–dimensional views and projections of bulk, shear, and Young's moduli. The degree of anisotropy of elastic modulus are as follows: β′–Mg7Tb > β′–Mg7Y > β′–Mg7Gd > β′–Mg7Dy. The density of states and electron localization function indicate that strong metallic bonds forming between RE atoms and their neighbouring Mg atoms make contributions to their better mechanical properties. The Abovementioned results will provide beneficial data references for the Mg alloy design and characterization.

Using an Ecological Approach to Explore Teacher Agency during the Implementation of a Citizen Science Education Program Using Arduino

Abstract Citizen science education is a new approach in science education for promoting scientific inquiry related to localized problems and for engaging in social action based on inquiry results. Using agency as a lens for understanding teachers’ practices when using this approach is important. In this ethnographic case study, a teacher implementing a citizen science education program using Arduino was investigated from an ecological approach using temporal and relational dimensions of agency. In the iterational dimension, the teacher’s own experiences and traits from life and professional histories were identified. His identity as a teacher and his religious values formed the projective dimension. Encouraging administrators, the financial difficulties of the school, and the COVID-19 pandemic were major elements of the practical-evaluative dimension. Findings reveal the complex array of the teacher’s agency in the context of implementing citizen science education with Arduino with students, which contributes new understandings about science teacher agency.

Having less: A personal project taxonomy of consumers' decluttering orientations, motives and emotions

AbstractThis study examines consumers' orientations toward decluttering and what differentiates consumers' engagement with decluttering as a routine social practice. An internet survey was conducted with a convenience sample of 172 adult consumers who recently engaged in home decluttering projects. Cluster analysis used appraisals of personal project dimensions to create a taxonomy in which we identified three orientations toward decluttering: disengaged, enthusiastic, and challenged. Subsequent analyses compared appraisals of success and affective and motivational differences across the three orientations. Findings reveal that decluttering is enjoyable, easy, and most successful for consumers with an enthusiastic orientation who intrinsically value decluttering and link it to self‐identify. Those who lack time or adequate control experience more stress and less success. Circumstances, and especially normative pressures, seemed to drive these orientations. All orientations experienced relatively high positive emotions after decluttering, which substantiates the positive influence of decluttering on emotional well‐being.

Mining Manchuria's Colonial Past

AbstractThe rise of nationalism in China in recent decades has grown in part out of a state-sponsored patriotic education project centered on the production of narratives involving the century-long history of China's subjection to foreign imperialism. In the northeast, the regional history of Russian and Japanese colonialism during the first half of the twentieth century has provided abundant materials for this collective commemoration of China's anti-imperialist struggles. A network of commemorative projects undertaken at local, regional, and national levels of the PRC state since the 1980s, including oral histories, monuments, museums, gazetteers, and party histories, has reframed the history of the northeast borderland in terms that legitimize post-Mao nationalist and market reform ideology. This study brings attention to another dimension of these commemorative projects, the translation and republication of Chinese and Japanese historical accounts originally produced in northern Manchuria during the 1920s and 1930s. I argue that the diverse representations of the borderland's history that are included in these accounts, while originally embedded in the colonial institutional contexts of Manchuria, acquired new significance in post-Mao reimaginings of the region's place in defining nationalism and negotiating Sino-Russian relations.

Characterization of eventually periodic modules in the singularity categories

The singularity category of a ring makes only the modules of finite projective dimension vanish among the modules, so that the singularity category is expected to characterize a homological property of modules of infinite projective dimension. In this paper, among such modules, we deal with eventually periodic modules over a left artin ring, and, as our main result, we characterize them in terms of morphisms in the singularity category. As applications, we first prove that, for the class of finite dimensional algebras over a field , being eventually periodic is preserved under singular equivalence of Morita type with level. Moreover, we determine which finite dimensional connected Nakayama algebras are eventually periodic when the ground field is algebraically closed.

Latent space projection of brain FDG‐PET creates a powerful classifier for neurodegenerative diseases

AbstractBackgroundFluorodeoxyglucose positron emission tomography (FDG‐PET) is an established modality for differential diagnosis of dementia. Deriving phenotypic signatures from FDG‐PET via machine learning is challenging due to the high dimensional nature of brain images relative to the generally small number of labeled examples available for training, the class imbalance among those labels, and the cooccurrence of multiple pathologies. In this study, we developed a multi‐class, multi‐label framework to address these challenges.MethodA database of clinically acquired PET/CT images from 3,000 unique patients was used to develop a latent space model using matrix decomposition. This model was then applied to images from a separate cohort of Mayo Clinic Alzheimer’s Disease Research Center participants (n=1,745) labeled as cognitively unimpaired (CU) (n=1,436) or with the following potentially co‐occurring phenotypes: Alzheimer’s disease (AD) (n=165), Lewy body dementia (DLB) (n=92), behavioral variant frontotemporal dementia (bvFTD) (n=43), semantic (svPPA) (n=10) and logopenic (lvPPA) (n=13) variant PPA, and posterior cortical atrophy (PCA) (n=17). A k‐nearest neighbors classifier that is robust to these imbalanced and overlapping labels was then trained on these examples. The resulting classifier was evaluated by area under receiver‐operator characteristic curve (ROC‐AUC) via leave one out cross validation, using clinical diagnosis as the gold standard.ResultROC curves and AUC scores for each phenotype are illustrated in Fig. 1a. Because the classifier is based on a k‐nearest neighbors connectivity matrix, it has a convenient graphical representation, where images are nodes and edges are drawn between an image and its set of nearest neighbors in latent space. A self‐organizing force directed graph constructed in this way is illustrated in Fig 1b, highlighting the strong separation of CU and degenerative images, as well as the segregation of each phenotype within the neurodegenerative region of the graph.ConclusionIn this study, we developed a machine learning framework for classification of neurodegenerative disease based on k‐nearest neighbor analysis in a low dimensional latent space projection of FDG‐PET images. By leveraging low‐dimensional representations and k‐nearest neighbors analysis, this framework is robust in multi‐class, multi‐label tasks with strong class imbalance and provides a highly interpretable graphical representation.

Latent space projection of brain FDG‐PET creates a powerful classifier for neurodegenerative diseases

AbstractBackgroundFluorodeoxyglucose positron emission tomography (FDG‐PET) is an established modality for differential diagnosis of dementia. Deriving phenotypic signatures from FDG‐PET via machine learning is challenging due to the high dimensional nature of brain images relative to the generally small number of labeled examples available for training, the class imbalance among those labels, and the cooccurrence of multiple pathologies. In this study, we developed a multi‐class, multi‐label framework to address these challenges.MethodA database of clinically acquired PET/CT images from 3,000 unique patients was used to develop a latent space model using matrix decomposition. This model was then applied to images from a separate cohort of Mayo Clinic Alzheimer’s Disease Research Center participants (n=1,745) labeled as cognitively unimpaired (CU) (n=1,436) or with the following potentially co‐occurring phenotypes: Alzheimer’s disease (AD) (n=165), Lewy body dementia (DLB) (n=92), behavioral variant frontotemporal dementia (bvFTD) (n=43), semantic (svPPA) (n=10) and logopenic (lvPPA) (n=13) variant PPA, and posterior cortical atrophy (PCA) (n=17). A k‐nearest neighbors classifier that is robust to these imbalanced and overlapping labels was then trained on these examples. The resulting classifier was evaluated by area under receiver‐operator characteristic curve (ROC‐AUC) via leave one out cross validation, using clinical diagnosis as the gold standard.ResultROC curves and AUC scores for each phenotype are illustrated in Fig. 1a. Because the classifier is based on a k‐nearest neighbors connectivity matrix, it has a convenient graphical representation, where images are nodes and edges are drawn between an image and its set of nearest neighbors in latent space. A self‐organizing force directed graph constructed in this way is illustrated in Fig 1b, highlighting the strong separation of CU and degenerative images, as well as the segregation of each phenotype within the neurodegenerative region of the graph.ConclusionIn this study, we developed a machine learning framework for classification of neurodegenerative disease based on k‐nearest neighbor analysis in a low dimensional latent space projection of FDG‐PET images. By leveraging low‐dimensional representations and k‐nearest neighbors analysis, this framework is robust in multi‐class, multi‐label tasks with strong class imbalance and provides a highly interpretable graphical representation.

Translating God on the Borders of Sovereignty

Abstract Over the long nineteenth century, translations of Christian scripture into indigenous languages were produced at a far greater rate than at any time previously, a product of both the rise of the modern Protestant missionary movement and the acceleration of British imperial and Anglo settler colonial conquest. This article explores a dimension of the global evangelical translation project: the translation of the word “God.” Where, at the close of the sixteenth century, there were just under 30 different words used to translate “God” in published vernacular, between 1800 and the early twentieth century close to 400 new “Gods” entered the Christian lexicon. Reading translation conflicts for what they tell us about the way power was imagined on the modern Anglo-American colonial frontier, this article argues that contests over the translation of “God” offer a window into the cultural and intellectual dimension of colonial conflict, and reveal a neglected chapter in the conceptual history of sovereignty: at the very moment when the concept of sovereignty was increasingly imagined on the model of the law-bound territorial state, alternative theories of sovereign power entered global circulation.

Project Successful Deployment

In this paper, we present a method for evaluating the success of digitalization projects, namely the Project Successful Deployment (PSD). With the term digitalization, we mean the use of digital technologies and digitized information to create value in new ways and to benefit from them. The existing methods for project evaluation emphasize the capability of a project to deliver its results by respecting times and costs. The method we propose, instead, suggests evaluating projects by means of its external dimensions, namely the functionalities and quality of the deliverables. These external project dimensions are reflected on the project scope, and thus evaluate the requirements of the deliverables, and the degree to which the deliverable meet their quality objectives The method is composed of a set of matrixes, and it uses a structured procedure to define and refine its items and their weights, by means of a panel of experts. It has been applied to a practical case study, a digitalization project of a network of research and teaching laboratories. The method allowed a structured project evaluation, and the practical case study showed strengths and weaknesses of the PSD model, which proved to be robust and effective, in providing a timely evaluation of the project.

Standardly stratified lower triangular [formula omitted]-algebras with enough idempotents

In this paper we study the lower triangular matrix K-algebra Λ:=[T0MU], where U and T are basic K-algebras with enough idempotents and M is an U-T-bimodule where K acts centrally. Moreover, we characterise in terms of U, T and M when, on one hand, the lower triangular matrix K-algebra Λ is standardly stratified in the sense of [15]; and on the other hand, when Λ is locally bounded in the sense of Gabriel [10]. Finally, we also study several properties relating the projective dimensions in the categories of finitely generated modules mod(U), mod(T) and mod(Λ).

Technology application in the mathematical learning

The teaching of mathematics these days can turn multidisciplinary with a great and significant content when making use of technology and project-based learning. This work arises as an answer of the Comprehensive Reform in the Development of Competencies petition, the proposal in the educational improvement by the constructive lineup of John Biggs and Collis, as well as Marzano´s use of significant knowledge and mental habits. The implementation of this engineering project was with teams of students from the first semester of different engineering majors in the subjects of algebra, geometry, and trigonometry. The evaluation of collaborative work in the classroom and computer room about mathematic laws, calculations, and mathematical software, as well as the reliable investigation on internet about the context and dimension of projects resulted in the creation of the first project contest, where every student showed their motivation and excellent development, the interaction of technology use and the mathematical learning.

Managerial Dimensions Of The Educational Intervention Projects

Managerial Dimensions Of The Educational Intervention Projects

On Yuzvinsky’s lattice sheaf cohomology for hyperplane arrangements

We establish the relationship between the cohomology of a certain sheaf on the intersection lattice of a hyperplane arrangement introduced by Yuzvinsky and the cohomology of the coherent sheaf on punctured affine space, respectively projective space associated to the module of logarithmic vector fields along the arrangement. Our main result gives a K\"unneth formula connecting the cohomology theories, answering a question by Yoshinaga. This, in turn, provides a characterization of the projective dimension of the module of logarithmic vector fields and yields a new proof of Yuzvinsky's freeness criterion. Furthermore, our approach affords a new formulation of Terao's freeness conjecture and a more general problem.

Solution to a conjecture on edge rings with 2-linear resolutions

ABSTRACT For a graph the edge ring is , where and I(G) is generated by . The conjecture we treat is the following. Conjecture 1. If has a 2-linear resolution, then the projective dimension of , pd , equals the maximal degree of a vertex in G. As far as we know, this conjecture is first mentioned in a paper by Gitler and Valencia [7, Conjecture 4.13], and there it is called the Eliahou-Villarreal conjecture. The conjecture is treated in a recent paper by Ahmed, Mafi, and Namiq [1]. That there are counterexamples was noted already by Moradi and Kiani [9]. By interpreting as a Stanley-Reisner ring, we are able to characterize those graphs for which the conjecture holds.

Homological invariants of unbounded complexes under Frobenius extensions

Let be a Frobenius extension of rings and M an S-complex. It is proven that if the Gorenstein projective (injective) dimension of M as an S-complex is finite, then the Gorenstein projective (injective) dimension of M as an S-complex equals the Gorenstein projective (injective) dimension of M as an R-complex. As a corollary one has that if the projective (injective) dimension of M as an S-complex is finite, then the projective (injective) dimension of M as an S-complex equals the projective (injective) dimension of M as an R-complex. This statement extends a known result of Nakayama and Tsuzuku.

Vanishing of local cohomology and set-theoretically Cohen–Macaulay ideals

In this paper, first, we generalize a result of Peskine–Szpiro on the relation between the cohomological dimension and projective dimension. Then, we give conditions for the vanishing of local cohomology from local to global and vice versa. Our final goal in the present paper is to examine the set-theoretically Cohen–Macaulay ideals to find some cohomological characterization of these kinds of ideals.

A non-linear version of Bourgain’s projection theorem

We prove a version of Bourgain’s projection theorem for parametrized families of C^2 maps, which refines the original statement even in the linear case by requiring non-concentration only at a single natural scale. As one application, we show that if A is a Borel set of Hausdorff dimension close to 1 in \mathbb{R}^2 or close to 3/2 in \mathbb{R}^3 , then for y\in A outside of a very sparse set, the pinned distance set \{|x-y|:x\in A\} has Hausdorff dimension at least 1/2+c , where c is universal. Furthermore, the same holds if the distances are taken with respect to a C^2 norm of positive Gaussian curvature. As further applications, we obtain new bounds on the dimensions of spherical projections, and an improvement over the trivial estimate for incidences between \delta -balls and \delta -neighborhoods of curves in the plane, under fairly general assumptions. The proofs depend on a new multiscale decomposition of measures into “Frostman pieces” that may be of independent interest.

Classes of cut ideals and their Betti numbers

We study monomial cut ideals associated to a graph G, which are a monomial analogue of toric cut ideals as introduced by Sturmfels and Sullivant. Primary decompositions, projective dimensions, and Castelnuovo–Mumford regularities are investigated if the graph can be decomposed as 0-clique sums and disjoint union of subgraphs. The total Betti numbers of a cycle are computed. Moreover, we classify all Freiman ideals among monomial cut ideals.

Australia’s Energy Security and Statecraft in an Era of Strategic Competition

Previously published studies on Australia’s energy security did not examine the country’s domestic energy security situation in tandem with its international energy statecraft. This paper fills this research gap. In order to provide a robust analysis of a country’s strategic options in the energy sector, it is paramount to balance domestic and international dimensions, along with internal and external factors. The paper utilizes the strengths, weaknesses, opportunities, and threats (SWOT) method to review Australia’s strategic options by balancing the inward-looking, domestic risk minimization dimension (energy security), with the outward-looking, international power projection dimension (energy statecraft). The paper also applies the 4A framework (availability, accessibility, affordability, and acceptance) to assess Australia’s energy security performance. On the one hand, the results demonstrate that Australia has many strengths and opportunities as a reliable and stable energy supplier, endowed with traditional and renewable energy resources, and critical minerals. On the other hand, numerous internal weaknesses and external threats may affect Australia’s strategic options in the future. The most pressing issue is the historical lack of strategic government intervention in the energy market, which has, paradoxically, resulted in domestic energy accessibility and affordability crisis. The market-based approach is also the main reason why Australia has not transformed its energy resources into capabilities to be used as instruments of statecraft. The paper uses the SWOT analysis and the 4A assessment as the basis for discussion on how Australia can transform its energy sector weaknesses and threats into strengths and opportunities, to benefit the national interest.