Terms Of Norms Research Articles

Optimal Rates of Approximation by Shallow ReLU$$^k$$ Neural Networks and Applications to Nonparametric Regression

AbstractWe study the approximation capacity of some variation spaces corresponding to shallow ReLU$$^k$$ k neural networks. It is shown that sufficiently smooth functions are contained in these spaces with finite variation norms. For functions with less smoothness, the approximation rates in terms of the variation norm are established. Using these results, we are able to prove the optimal approximation rates in terms of the number of neurons for shallow ReLU$$^k$$ k neural networks. It is also shown how these results can be used to derive approximation bounds for deep neural networks and convolutional neural networks (CNNs). As applications, we study convergence rates for nonparametric regression using three ReLU neural network models: shallow neural network, over-parameterized neural network, and CNN. In particular, we show that shallow neural networks can achieve the minimax optimal rates for learning Hölder functions, which complements recent results for deep neural networks. It is also proven that over-parameterized (deep or shallow) neural networks can achieve nearly optimal rates for nonparametric regression.

Typical Lipschitz images of rectifiable metric spaces

Abstract This article studies typical 1-Lipschitz images of 𝑛-rectifiable metric spaces 𝐸 into R m \mathbb{R}^{m} for m ≥ n m\geq n . For example, if E ⊂ R k E\subset\mathbb{R}^{k} , we show that the Jacobian of such a typical 1-Lipschitz map equals 1 H n \mathcal{H}^{n} -almost everywhere and, if m > n m>n , preserves the Hausdorff measure of 𝐸. In general, we provide sufficient conditions, in terms of the tangent norms of 𝐸, for when a typical 1-Lipschitz map preserves the Hausdorff measure of 𝐸, up to some constant multiple. Almost optimal results for strongly 𝑛-rectifiable metric spaces are obtained. On the other hand, for any norm | ⋅ | \lvert\,{\cdot}\,\rvert on R m \mathbb{R}^{m} , we show that, in the space of 1-Lipschitz functions from ( [ − 1 , 1 ] n , | ⋅ | ∞ ) ([-1,1]^{n},\lvert\,{\cdot}\,\rvert_{\infty}) to ( R m , | ⋅ | ) (\mathbb{R}^{m},\lvert\,{\cdot}\,\rvert) , the H n \mathcal{H}^{n} -measure of a typical image is not bounded below by any Δ > 0 \Delta>0 .

Building coercive Lyapunov–Krasovskii functionals based on Razumikhin and Halanay approaches

AbstractIn this article, we provide a systematic and constructive way to build a Lyapunov–Krasovskii functional for time‐delay systems whose stability can be established through the Razumikhin or the Halanay approaches. The constructed Lyapunov–Krasovskii functional turns out to be coercive, meaning sandwiched between functions of the state history norm, and to dissipate in terms of the whole history norm. We present these results in the framework of input‐to‐state stability (ISS) in order to further account for the influence of input disturbances. A special emphasis is also given on exponential stability and exponential ISS. We illustrate our findings though the study of a coupled ODE‐PDE model of a chemical reactor, and show that, unlike most results in that area, our approach happens to ensure ISS in terms of the supremum norm of the state.

Quantitative evaluation of China's energy supply policies in the ‘13th Five‐Year Plan’ period (2016–2020): A PMC index modelling approach incorporating text mining

AbstractEnergy supply policy quality is an important factor that would impact national energy security. While existing research conducted policy evaluation from the lens of performance assessment, less study has been devoted to evaluating energy supply policy from the perspective of policy making. This study takes the energy supply policy documents issued by China's central government during the ‘13th Five‐Year Plan’ period (2016–2020) as the research sample, and pioneers the use of the extended policy modelling consistency (PMC) index model combined with the text mining methodology to construct a policy evaluation index system with the characteristics of the energy supply policy which conducts a more in‐depth quantitative evaluation of the energy supply policy documents. The results show that the average PMC index value of China's energy supply policies is 7.26, and the overall quality is high. The concerns of existing policies are mainly focused on reform and development planning, safety production management and project engineering construction, but there is still room for improvement in policy predictability, coordination of policy areas, clarity of policy basis and goals and comprehensiveness of policy tool combinations. Based on this, China should improve the comprehensiveness of its energy supply policy tool combinations in terms of policy design norm; strengthen policy predictability and coordination of policy areas in terms of policy implementation guarantee; and clarify the basis and goals of policies in terms of policy orientation enhancement, to provide a reference basis for the formulation and improvement of future energy supply policies and to realise the sustainable development of energy supply.

Estimates on the velocity of a rigid body moving in a fluid

We obtain estimates of all components of the velocity of a 3D rigid body moving in a viscous incompressible fluid without any symmetry restriction on the shape of the rigid body or the container. The estimates are in terms of suitable norms of the velocity field in a small domain of the fluid only, provided the distance h between the rigid body and the container is small. As a consequence we obtain suitable differential inequalities that control the distance h. The results are obtained using the fact that the vector field under investigation belongs to suitable function spaces, without any use of hydrodynamic equations.

Understanding the Religions of India

In India, in all communities, individuals have the right to practice the religion of his or her own choice. The religions have an important influence in the lives of the individuals. The individuals generate information in terms of norms and values through practicing their religion. In the lives of the individuals, in both personal and professional spheres, the practicing of religion is regarded as vital. In this manner, the individuals are able to benefit in number of ways, i.e. putting into operation all types of job duties successfully; inculcating the traits of morality and ethics; implementing the traits of diligence, resourcefulness and conscientiousness; developing mutual understanding with others and promoting good health, physically and psychologically. As a consequence, individuals are able to render an important contribution in promoting enrichment of their overall personality traits and standards of living. The individuals, belonging to all communities, neighbourhoods and professional settings need to accept the religions of other individuals. In other words, individuals need to form positive viewpoints in terms of all religions and accept these. One of the important aspects that needs to be taken into account is, one should not possess negative feelings in terms of any other religion. Therefore, it is understood on a comprehensive basis that religions are vital in promoting enrichment of overall quality of lives of individuals. The main concepts that are taken into account in this research paper are, religions of India, understanding the objectives of practicing religions and factors highlighting the importance of religion in India.

Peran Media Digital dalam Bingkai Etnopedagogik sebagai Upaya Optimalisasi Pencapaian Kualitas Pendidikan Anak Usia Dini Masa Depan

The digitalization program for early childhood education media in the era of the industrial revolution 4.0 is an opportunity as well as a challenge for educational inventors in the world. This opportunity can be used as an exploratory land for the development of science for educational media inventors. On the other hand, the challenges are related to the clash of value systems and consistency in strengthening the integration of the conservation function system with the educational innovation function. In this context, challenge does not mean weakness, but for scholars it is a form of motivational strength for the practical development of knowledge to be used as material for increasing self-capacity as a true scientist. Thus, the opportunities and challenges that arise in this study become a source of energy to overcome anxiety about the future of education. With regard to ethno-pedagogy which is the contextual realization of human imagination in terms of human values ​​and norms, it has a very important role in presenting every digital product that is created. So that the process of integrating conservation values ​​with educational innovation values ​​can show logical consequences for achieving the expected quality of early childhood education in the future.

SMES’ ENERGY EFFICIENCY IMPROVEMENT: THE ROLES OF KNOWLEDGE, GOVERNMENT INTERVENTION AND PERSONAL NORMS

Energy efficiency is like the ‘low hanging fruit’ in which improving energy efficiency is one of the easiest ways to lessen energy demand. Firms, large or small, can make a major positive contribution to energy sustainability by improving energy efficiency. This research attempts to examine the effects of SMEs’ perceptions in terms of knowledge, government intervention and personal norms on their support for energy efficiency improvement. A survey was carried out to collect data and empirical testing of the research model was conducted. The estimated results suggest that the factors of knowledge and government intervention have significant effects on SMEs’ support for energy efficiency improvement. In addition, the results show that personal norms is an important mediator variable in influencing SMEs’ decision to improve energy efficiency. The findings of this research provide some fresh insights on energy efficiency improvement of SMEs and several notable implications can be drawn from this research.

Трансформации социальной структуры и политических институтов традиционного черкесского общества через призму хабзэ в конце XVIII – начале XIX в.

The article is devoted to the analysis of transcripts of Circassian adats / khabze in terms of norms relating to class relations and political institutions (pshchyshkhue / grand duke / valii, hey / princely court, mekhkeme) of the late 18th - early 19th centuries. The norms of the period under review indicate a democratic transformation of the class structure of traditional Circassian society and the influence of Sharia on political institutions. Khabze, recorded in the 1840s and 1860s, demonstrate that the code of Adyghe khabze, despite the divergence of socio-political development in the east and west of the country, continued to develop as a single system of law.

Exploration of an Innovative Model of Higher Education Management and Student Training Mechanism Based on Cognitive Mapping

Abstract In this paper, we use cognitive mapping for four-parameter Logistics model fitting computation and obtain personalized learning resource paths incorporating learners’ fuzzy cognitive mapping after eliminating the influence of blunders and guessing factors on learners’ cognitive assessment. After calculating peer similarity based on the cognitive map, the cognitive change trajectory is tracked, and learning resources are recommended iteratively until students become proficient. Based on the cognitive objective function and knowledge relevance, the cognitive mapping of education management of college students is constructed. The research on the mechanism of higher education management and student cultivation is analyzed from the goal of talent cultivation in colleges and universities. Statistical analysis is utilized to analyze the research on the mechanism of higher education management and student cultivation. The results show that the t-value of the regression results are all greater than 2, the regression results are significant, and its regression coefficient is 0.87, which means that all dimensions of the cultivation mechanism of colleges and universities have a positive impact on external employability. In terms of academic attitude, moral cultivation and academic norms, the supervisor has a greater role in influencing postgraduates, and its percentage values are 49.2%, 52.1% and 46.5%, respectively, and the mean values of the three dimensions are above 4.1, while the standard deviation values are above 1.0. This study provides basic references and effective suggestions to promote the continuous improvement of graduate student training quality and is of great significance to the innovation of educational management and student training mechanisms.

A duality approach to regularized learning problems in Banach spaces

Regularized learning problems in Banach spaces, which often minimize the sum of a data fidelity term in one Banach norm and a regularization term in another Banach norm, is challenging to solve. We construct a direct sum space based on the Banach spaces for the fidelity term and the regularization term, and recast the objective function as the norm of a quotient space of the direct sum space. We then express the original regularized problem as an optimization problem in the dual space of the direct sum space. It is to find the maximum of a linear function on a convex polytope, which may be solved by linear programming. A solution of the original problem is then obtained by using related extremal properties of norming functionals from a solution of the dual problem. Numerical experiments demonstrate that the proposed duality approach is effective for solving the regularization learning problems.

ADAPTIVE REUSE OF HERITAGE ARCHITECTURE AS A SUSTAINABLE DEVELOPMENT TOOL: OPPORTUNITIES AND CHALLENGES

History plays an important role in defining our present and future. Architectural heritage is the symbol of our cultural identities, acting as remnants from the past. Respect and values of the architecture heritage affect the identity and vision of the culture. The existence of architecture is defined in both time and space, therefore the need for historical monuments to be adapted and reused suited for current time while satisfying the historical context and cultural values, is the present-day demand to bring people together who are otherwise separated by certain boundaries. With the current technological advancement, the 21st century skyline places focus on glass and metal facades that lack personality, hence there is a need for more effort to conserve and readapt heritage sites that stem from our cultures and our values. The purpose of the study is to create awareness towards a more sustainable approach for heritage conservation by adaptive reuse and understand the influence it has on the various sectors of a society and its residents. Based on a qualitative methodology, with a case study approach, this paper uses observation, in-depth interviews, and textual analysis to explore possibilities of sustainable transformation of architectural heritage in present and transferring the culture and values to future. This research paper will open further opportunities in this field to understand the factors behind the influence of these intangible and tangible variables and can be utilized to raise a country to an international standard of competition in terms of economy and societal norms.

Dual Discriminative Low-Rank Projection Learning for Robust Image Classification

Numerous methods have exploited projection learning to extract low-dimensional features for image classification. Some projection learning methods integrate low-rank matrix recovery into classification models to equip the projection subspace with discrimination and robustness against corruption. However, these methods cannot directly recover “clean” components from the new datum in a low-dimensional subspace. Additionally, they are sensitive to the selection of projection dimensions. To overcome these shortcomings, we propose a dual discriminative low-rank projection learning framework for robust image classification. Specifically, the proposed method learns a low-rank projection and a semi-orthogonal projection to recover “clean” components from the original data and simultaneously obtain a low-dimensional subspace. Thereafter, to preserve discriminative information in the low-dimensional subspace, an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2,1</sub> -norm term is constructed by concentrating the projected intra-class samples around their adaptive class centroids. Regression-based terms are appended using the low-dimensional features extracted from the recovered clean data and the class centroids for more accurate classification. Experiments on five public databases with various corruptions demonstrate that the proposed method can robustly classify image data despite a small training sample sizes and gross corruption. The superiority of the proposed method is further verified on the large-scale PubFig83 database, on which it achieves an 87.58% classification accuracy.

Pelaksanaan Tanggung Jawab Nakhoda Kapal atas Kerusakan Kapal dalam Pengangkutan Penumpang Antar Wilayah Kabupaten Natuna di Pelabuhan Rakyat Midai

Purpose: There is a problem that occurred in Natuna Regency where a ship experienced engine damage, causing the ship to come to a halt in the middle of the sea and leaving passengers stranded at sea for an extended period. This situation can be hazardous and poses a risk to maritime safety. During disasters such as ship sinking, grounding, or fires, the risk to passenger safety increases. The high demand for facilities and infrastructure in passenger transportation should be a top priority. This research was conducted with the aim of determining the captain's responsibility for passenger safety to prevent any negligence on the part of the skipper. Methodology/approach: The Normative-Empirical Law research method involves understanding law in terms of norms (rules) and the application of legal rules in actual behavior resulting from the enforcement of legal norms. The captain is required to adhere to all applicable rules and regu lations to ensure the ship's navigability, safety of the vessel, passenger safety, and cargo. The skipper is not permitted to embark on a journey unless the ship is adequately equipped and prepared, as referred to in Article 343 of the Criminal Code. Results/findings: Regarding the implementation of the captain's responsibility for ship damage during passenger transportation within the Natuna Regency area, the captain pays more attention to the operational conditions of the ship to avoid issues related to ship damage. Additionally, the sub-district head of Midai is advised to be more selective in transferring responsibility to the skipper of the KM Fisabilillah 1 ship.

Studi Komparatif Theory Planned Behavior antara yang Sudah dan yang Belum Tersertifikasi Halal di Sumenep

This study aims to determine the comparison of Theory Planned Behavior of MSME actors between those who have and those who have not been halal certified in Sumenep. The method used is a qualitative descriptive approach, data collection is obtained from interviews, observation, and documentation. And the technique of taking informants using purposive sampling technique. The results showed that in terms of the attitude of the reasons why Sumenep MSME players want to carry out halal certification is because they consider that halal certification can have a positive impact on sales. Meanwhile, MSME players who do not have halal certification consider that halal certification has no impact on sales. Then, from the subjective norm factor that encourages MSME players to want to carry out halal certification due to requests and support from people in the surrounding environment. Meanwhile, MSME players who do not yet have halal certification, they do not get support or requests from their surroundings. In terms of perceived behavioral control, the reason why MSME players want to carry out halal certification is due to their perception of the ease of conducting halal certification. Meanwhile, MSME players who do not yet have halal certification consider that the process of applying for halal certification is difficult. So it can be concluded that the behavior of MSME players Sumenep is good in terms of attitudes, subjective norms and perceptions of contradictions.

Membedah Tindak Pidana Politik Uang, Suatu Telaah Dari Sisi Struktur Norma

Introduction: Money politics is one of the crimes regulated in the Election Law. This crime in the Election Law regime is considered a Structured, Systematic and Massive Crime (TSM). This crime has always been a discourse that surfaced in the public when approaching elections / elections. Especially now, Indonesia is undergoing a political year leading to the upcoming 2024 elections. &#x0D; Purpose of the Research: this paper aims to dissect the criminal act of money politics in terms of the norm structure. &#x0D; Method of the Research: this paper uses the method of statutory approach (statute approach), conceptual approach (conceptual approach) and philosophical approach to legal (philosophical approach). &#x0D; Results of the Research: norm structure can be divided into 4 (four) elements, namely norm subjects (normadressat), norm operators (modus van behoren), norm objects (normgedrag) and norm conditions (normconditie). The criminal act of money politics mentioned in this paper only revolves around the norms regulated in article 515 and article 519 of the Election Law. Basically the two articles have similarities in the subject elements of norms and norm operators. The subject of the norm of both articles is "everyone", while the operator of the norm is "prohibition". The two articles differ in terms of objects, norms and norm conditions. For article 515 the object of the norm is the act of "promising or giving money or other material. The norm condition is "intentionally, voting, to the Voter not to exercise his right to vote, to vote for certain Election Participants, to exercise his right to vote in a certain way, the ballot is invalid". As for article 519 the object of the norm is "fraudulent acts", and the norm condition is "misleading someone, coercing, promising or by giving money or other materials, to obtain support for the candidacy of DPD members in elections".

Precise eigenvalues in the solutions of generalized Sturm–Liouville problems

In the paper, we determine the eigenvalues and eigenfunctions of the generalized Sturm–Liouville problems, whose potentials may be nonlinear functions of eigen-parameter, by developing new iterative algorithms based on the fictitious time integration method (FTIM) and half-interval method (HIM). We derive two eigen-parameter dependent linear shape functions, from which we can transform the generalized Sturm–Liouville problem to an initial value problem for a new variable, and automatically preserve the prescribed eigen-parameter dependent Sturm–Liouville boundary conditions. A nonlinear equation in terms of the relative norm of two consecutive right-end values of the new variable is derived for iteratively determining the eigenvalue by using the FTIM. The resultant sequence of the iterated eigenvalues are monotonically convergent to the desired eigenvalue, and meanwhile the unknown initial values of the eigenfunction can be determined for computing the eigenfunction by integrating the Sturm–Liouville equation. We propose a high precision point target method by using the HIM to determine the eigenvalue and eigenfunction of the generalized Sturm–Liouville problem, which is transformed to a definite initial value problem with a simpler Dirichlet or Neumann boundary condition on the right-end as a point target equation. Several new theoretical results are proved such that more simple generalized Sturm–Liouville problem can be derived with a single point target on the right-end. Depending on different transformation techniques and the derived target equations, seven types FTIM and four types HIM are developed. Numerical examples confirm that the present FTIM and HIM are effective and accurate.

How does herd behaviour impact the purchase intention? Explore the moderating effect of risk aversion in the context of Vietnamese consumers

Managers have been applying the influence of herd behaviour to stimulate purchase intention but have not achieved precise results because they do not clearly understand the mechanism of this relationship. This article aims to determine the effect of herd behaviour on the relationship between information adoption and purchase intention and the moderating of risk aversion. With 17 variables for four concepts, data was collected online with 340 respondents and analysed with a partial least squares structural equation modelling. When considering herd behaviour in terms of subjective norms with corroborated information, this study upgrades the classic role of the theory of planned behaviour in purchase intention. The findings indicate a positive relationship between herd behaviour, information adoption and purchase intention. In particular, risk aversion is a positive moderator of the relationship between herd behaviour and purchase intention. This result is a significant addition to the theoretical body and shows that risk aversion can drive purchase intention through herd behaviour. In other words, this study upgrades the perception of information source authentication through herd behaviour to explain the purchase intentions of Vietnamese consumers. These results show that managers can create risky situations to promote herd behaviour. They can also apply information adoption stimuli to stimulate herd behaviour. At the same time, through the information adoption mechanism, consumers can be more alert to the effect of herd behaviour and have more opinions when making purchases.

Exploring challenges in later-life relationships: a qualitative study of the queries posted on counselling websites.

Divorce rates among individuals aged 50 and above are on the rise. Given the greater life expectancy compared to previous generations, this is an issue that is affecting an increasing number of people. Therefore, it warrants an inquiry into the challenges these individuals encounter in their intimate relationships. This study analyzed 225 relationship-related queries posted on Czech counselling websites to identify the strains and stressor patterns that older adults face in their relationships. The queries were limited to those that concerned themes and problems related to partnerships, were posted by one of the partners aged 60 or over, and were analyzed using inductive thematic analysis. Four main relationship issues were identified: infidelity and jealousy; relationship estrangement and cooling; undesirable changes in personality; and illness and somatic issues. Additionally, three recurring themes were identified that made the problems more demanding and that were specific to older age: lack of norms for relationships in that age group, absence of resources to tackle the issues, and personal calculation for Time Spent and Time Remaining. The research found that the types of problems encountered by older adults were similar to those experienced by younger individuals. However, the way these problems were perceived and processed was influenced by specific aspects of aging, such as societal expectations, available coping resources, and the perception of time. The findings also highlighted the challenges faced by older adults in terms of relationship norms, sexual functioning, and personality changes.

On the abc$abc$ conjecture in algebraic number fields

AbstractIn this paper, we prove a weak form of the conjecture generalised to algebraic number fields. Given integers satisfying , Stewart and Yu were able to give an exponential bound in terms of the radical over the integers (Stewart and Yu [Math. Ann. 291 (1991), 225–230], Stewart and Yu [Duke Math. J. 108 (2001), no. 1, 169–181]), whereas Győry was able to give an exponential bound in the algebraic number field case for the projective height in terms of the radical for algebraic numbers (Győry [Acta Arith. 133 (2008), 281–295]). We generalise Stewart and Yu's method to give an improvement on Győry's bound for algebraic integers over the Hilbert Class Field of the initial number field K. Given algebraic integers in a number field K satisfying , we give an upper bound for the logarithm of the projective height in terms of norms of prime ideals dividing , where L is the Hilbert Class Field of K. In many cases, this allows us to give a bound in terms of the modified radical as given by Masser (Proc. Amer. Math. Soc. 130 (2002), no. 11, 3141–3150). Furthermore, by employing a recent bound of Győry (Publ. Math. Debrecen 94 (2019), 507–526) on the solutions of S‐unit equations, our estimates imply the upper bound where is an effectively computable constant. Further, given conditions on the largest prime ideal dividing , we obtain a sub‐exponential bound for in terms of the radical. Independently, as a direct application of his bounds on the solutions of S‐unit equations(Győry ([Publ. Math. Debrecen 94 (2019), 507–526]), Győry (Publ. Math. Debrecen 100 (2022), 499–511) also attains results mentioned above, including the above inequality, but over the base field K, as discussed in Section 6. As a consequence of our results, we will give an application to the effective Skolem–Mahler–Lech problem and give an improvement to a result by Lagarias and Soundararajan (J. Théor. Nombres Bordeaux 23 (2011), no. 1, 209–234) on the XYZ conjecture.