Dual Space Research Articles

Existence Results for a Dirichlet Boundary Value Problem Involving the p(x)-Laplacian Operator

The aim of this paper is to establish the existence of weak solutions, in$W_0^{1,p(x)}(\Omega)$, for a Dirichlet boundary value problem involving the$p(x)$-Laplacian operator. Our technical approach is based on the Berkovits topologicaldegree theory for a class of demicontinuous operators of generalized $(S_+)$ type. Wealso use as a necessary tool the properties of variable Lebesgue and Sobolev spaces, andspecially properties of $p(x)$-Laplacian operator. In order to usethis theory, we will transform our problem into an abstractHammerstein equation of the form $v+S\circ Tv=0$ in the reflexiveBanach space $W^{-1,p'(x)}(\Omega)$ which is the dual space of$W_0^{1,p(x)}(\Omega)$. Note also that the problem can be seen as anonlinear eigenvalue problem of the form$Au=\lambda u,$ where$Au:=-\Div(|\nabla u|^{p(x)-2}\nabla u)-f(x,u)$. When this problemadmits a~non-zero weak solution $u$, $\lambda$ is an eigenvalue ofit and $u$ is an associated eigenfunction.

Performing encounters (and encountering performance) at Auckland Zoo

Performing encounters (and encountering performance) at Auckland Zoo

A note on p-limited sets in dual Banach spaces

A note on p-limited sets in dual Banach spaces

Study on Legal Issues of Ocean Fishing in China

“The 21st century is the century of the ocean.” In recent years, China has paid more and more attention to the promotion and development of the marine industry, especially the ocean fishery has brought immeasurable economic benefits to China. The development of the marine field is becoming more and more important in the national political, economic and cultural development. All coastal countries have included marine development in their national development strategies and continuously improved their marine legislation under the provisions of the United Nations Convention on the law of the sea. China’s pelagic fishing began in 1985, but because China’s pelagic fishing started too late, after the entry into force of the United Nations Convention on the law of the sea at the end of 1994, nearly 36% of the richest high seas on earth became the exclusive economic zone of coastal countries, and the development space of China’s pelagic fishing has become very limited. After just more than 30 years of development, ***. However, before that, China’s pelagic fishing was still subject to the dual norms of international conventions and domestic laws, and China had not yet formulated a special law on pelagic fishing, and there were still many deficiencies in the legal system norms of pelagic fishing. Therefore, the biggest problem facing China’s pelagic fishery is how to better develop the marine industry under the system of laws and regulations, drive the coordinated economic development, provide legal guidance and help for pelagic fishermen, and provide solid technical support for building a marine power with Chinese characteristics.

Non-unital operator systems that are dual spaces

We will extend the main result of [1] to the non-unital case with a totally different proof. More precisely, we give an abstract characterization of an arbitrary self-adjoint weak⁎-closed subspace of L(H) (equipped with the induced matrix norm, the induced matrix cone and the induced weak⁎-topology). In order to do this, we give a matrix analogue of a result of Bonsall for ⁎-operator spaces equipped with closed matrix cones.

Semigroups for quadratic evolution equations acting on Shubin–Sobolev and Gelfand–Shilov spaces

We consider the initial value Cauchy problem for a class of evolution equations whose Hamiltonian is the Weyl quantization of a homogeneous quadratic form with non-negative definite real part. The solution semigroup is shown to be strongly continuous on several spaces: the Shubin-Sobolev spaces, the Schwartz space, the tempered distributions, the equal index Beurling type Gelfand-Shilov spaces and their dual ultradistribution spaces.

Approximations of Intuitionistic Fuzzy Ideals over Dual Spaces by Soft Binary Relations

The major advantage of this proposed work is to investigate roughness of intuitionistic fuzzy subsemigroups (RIFSs) by using soft relations. In this way, two sets of intuitionistic fuzzy (IF) soft subsemigroups, named lower approximation and upper approximation regarding aftersets and foresets, have been introduced. In RIFSs, incomplete and insufficient information is handled in decision-making problems like symptom diagnosis in medical science. In addition, this new technique is more effective as compared to the previous literature because we use intuitionistic fuzzy set (IFS) instead of fuzzy set (FS). Since the FS describes the membership degree only but often in real-world problems, we need the description of nonmembership degree. That is why an IFS is a more useful set due to its nonmembership degree and hesitation degree. The above technique is applied for left (right) ideals, interior ideals, and bi-ideals in the same manner as described for subsemigroups.

Climate action and growing electricity demand: Meeting both challenges in the 21st century with space-based solar power delivered by space elevator

Global climate action is the grand challenge of the 21st century. Large reductions in greenhouse gas emissions are needed, with net-zero CO2 emissions by mid-century a requirement for many scenarios that restrain global warming below 1.5 °C by 2100. This will require massive deployment of renewables such as ground-based solar and wind, increased long-distance grid transmission capacity, and electrification of much of the global economy. Increased electrification, coupled with rising standards of living in developing countries, will drive ever-larger electricity demand to 2050 and beyond. And the intermittency, land-use, and transmission requirements of ground-based solar and wind present challenges to large-scale deployment to meet Net Zero targets. Space-based solar power (SSP) presents an enticing alternative, and with full reusability of upcoming heavy launch vehicles and modular SSP architectures, the economics of deployment are expected to become favorable in the coming decades. Efforts exist today to deploy demonstrators this decade, with initial production systems by 2050. However, deploying SSP at the scale required to address rising, global electricity demand will itself require orders-of-magnitude greater launch frequencies. If heavy launch frequency does not increase quickly enough, due to environmental concerns, launch site competition, or other bottlenecks, the dropping cost of launch will not be sufficient to enable SSP for global impact this century. A space elevator (SE) development program like that proposed by the International Space Elevator Consortium (ISEC) offers an alternative delivery mechanism for scaling SSP beyond 2050, hedging the risk of reliance on a single delivery system while reducing the time required for large-scale SSP deployment. To de-risk possible futures in which SSP is a major part of the global energy mix this century, a dual space access architecture is proposed, leveraging the strengths of heavy launch and SEs together. Such an architecture would secure the future for green energy on Earth, in 2050 and beyond.

THE NON-DEGENERACY OF THE SKEW-SYMMETRIC BILINEAR FORM OF THE FINITE DIMENSIONAL REAL FROBENIUS LIE ALGEBRA

A Frobenius Lie algebra is recognized as the Lie algebra whose stabilizer at a Frobenius functional is trivial. This condition is equivalent to the existence of a skew-symmetric bilinear form which is non-degenerate. On the other hand, the Lie algebra is Frobenius as well if its orbit on the dual vector space is open. In this paper, we study the skew-symmetric bilinear form of finite dimensional Frobenius Lie algebra corresponding to its Frobenius functional. The work aims to prove that a Lie algebra of dimension is Frobenius if and only if the -th derivation of the Frobenius functional is not equal to zero. Indeed, this condition implies that the skew-symmetric bilinear form is non-degenerate and vice versa. In addition, some properties of Frobenius functionals are obtained. Furthermore, the computations are given using the coadjoint orbits and the structure matrix. As a discussion, we can investigate these results in the algebra case whether giving rise to a left-invariant K hler structure of a Frobenius Lie group or not.

The Smarandache Curves and Corresponding to Ruled Surfaces Due to N-Bishop Frame In Dual Space

Dual uzayda dual birim küre üzerinde seçilen dual Smarandache eğrisi Öklid-3 uzayındaki yönlü doğruların oluşturmuş olduğu regle yüzeye karşılık gelir. Bu çalışmada dual N-Bishop çatısının dual bileşenlerinin yardımıyla oluşturulan dual Smarandache eğrilerine karşılık gelen regle yüzeylerine ait bazı karakterizasyonlar incelenmiştir.

Anisotropic variable Campanato-type spaces and their Carleson measure characterizations

Let $$p(\cdot ):\ {\mathbb {R}^n}\rightarrow (0,\infty )$$ be a variable exponent function satisfying the globally log-Hölder continuous condition and A a general expansive matrix on $${\mathbb {R}^n}$$ . In this article, the authors introduce the anisotropic variable Campanato-type spaces and give some applications. Especially, using the known atomic and finite atomic characterizations of anisotropic variable Hardy space $$H_A^{p(\cdot )}(\mathbb {R}^n)$$ , the authors prove that this Campanato-type space is the appropriate dual space of $$H_A^{p(\cdot )}(\mathbb {R}^n)$$ with full range $$p(\cdot )$$ . As applications, the authors first deduce several equivalent characterizations of these Campanato-type spaces. Furthermore, the authors also introduce the anisotropic variable tent spaces and show their atomic decomposition. Combining this and the obtained dual theorem, the Carleson measure characterizations of these anisotropic variable Campanato-type spaces are established.

Dual of Modulation Spaces with Variable Smoothness and Integrability

In this article, we first give a proof for the denseness of the Schwartz class in the modulation spaces with variable smoothness and integrability. Then, we study the dual spaces of such modulation spaces.

Dual spaces for martingale Musielak–Orlicz Lorentz Hardy spaces

In this article, under the mild assumption that the Doob maximal operator is bounded on Musielak–Orlicz spaces, the authors first establish the atomic characterization of martingale Musielak–Orlicz Lorentz Hardy spaces. Using atomic characterizations, the authors then clarify the relations among five martingale Musielak–Orlicz Lorentz Hardy spaces and construct the generalized martingale BMO type spaces which prove to be the dual spaces of martingale Musielak–Orlicz Lorentz Hardy spaces. As applications, the authors further investigate John–Nirenberg inequalities by the dual method. A novelty of this article is to apply the assumption on the boundedness of the Doob maximal operator to develop a theory of martingale spaces and hence the obtained results have a wide generality.

Dual space vector PWM technique for a three-phase to five-phase quasi Z-source direct matrix converter

ABSTRACT This article proposes a dual space vector PWM control technique for a three-phase to five-phase quasi Z-source direct matrix converter (QZSDMC). The proposed circuit consists of a quasi-impedance source network along with a five-phase traditional matrix converter. With the observation of bidirectional operating capability and the utilization of shoot-through duty ratio, the proposed converter can ensure both buck and boost operations. This converter overcomes the voltage transfer ratio restriction and poor voltage regulation of the traditional matrix converter. Also, this converter ensures continuous input current with an absence of input LC filter components. The proposed direct AC to AC power converter gives an alternate solution to the multiphase drive system instead of the traditional voltage source inverter. The developed dual space vector PWM control technique for three-phase to five-phase voltage conversion is derived from the basic space vector PWM approach. The proposed dual space vector PWM technique utilizes 90 active states and 3 zero vector states among 243 switching states. The QZSDMC has been analysed with a dual space vector PWM approach along with the distribution of shoot-through state in the zero vector. The proposed topology along with the control strategy has been simulated using MATLAB/Simulink environment.

Analyzing the dual space of the saturated ideal of a regular set and the local multiplicities of its zeros

In this paper, a method is proposed to analyze the structure of dual space of the saturated ideal generated by a regular set and the local multiplicities of its zeros. In detail, we generalize the squarefree decomposition of univariate polynomials to the so‐called pseudo squarefree decomposition of multivariate polynomials and then propose an algorithm for decomposing a regular set into a finite number of simple sets. From the output of this algorithm, the multiplicities of zeros could be directly read out, and the real solution isolation with multiplicity can also be easily produced.

Generalized Whittaker Quotients of Schwartz Functions onG-Spaces

AbstractLet $G$ be a reductive group over a local field $F$ of characteristic zero. Let $X$ be a $G$-space. In this paper we study the existence of generalized Whittaker quotients for the space of Schwartz functions on $X$, considered as a representation of $G$. We show that the set of nilpotent elements of the dual space to the Lie algebra such that the corresponding generalized Whittaker quotient does not vanish contains the nilpotent part of the image of the moment map and lies in the closure of this image. This generalizes recent results of Prasad and Sakellaridis. Applying our theorems to symmetric pairs $(G,H)$ we show that there exists an infinite-dimensional $H$-distinguished representation of $G$ if and only if the real reductive group corresponding to the pair $(G,H)$ is non-compact. For quasi-split $G$ we also extend to the Archimedean case the theorem of Prasad stating that there exists a generic $H$-distinguished representation of $G$ if and only if the real reductive group corresponding to the pair $(G,H)$ is quasi-split. In the non-Archimedean case our result also gives rather sharp bounds on the wave-front sets of distinguished representations. Finally, we deduce a corollary on vanishing of period integrals of automorphic forms with certain Whittaker supports. This corollary, when combined with the restrictions on the Whittaker support of cuspidal automorphic representations proven by Gomez–Gourevitch–Sahi, implies many of the vanishing results on periods of automorphic forms proved by Ash–Ginzburg–Rallis.

Soft sensor of iron tailings grade based on froth image features for reverse flotation

In the iron reverse flotation production process, the amount of flotation agent and the quality of flotation products are usually judged according to the grade of tailings, so it is essential to measure the grade of tailings froth. This research applies computer vision and image feature extraction technology to the soft sensor of tailings froth grade. An adaptive selection method for the image target region is proposed. The relationship between RGB (Red, Green, Blue), HSI (Hue, Saturation, Intensity), and Lab color space and tailings grade of reverse flotation in iron mine has been analyzed. A new image feature is proposed to characterize the degree of froth mineralization. The RGB and HSI dual color space feature values and froth mineralization degree values are determined as input, and the tailing grade soft sensor model is established by the multilayer feedforward perceptrons and VGG-19 neural network. A tailings grade soft sensor system has been developed and applied in a flotation workshop. The results of industrial tests show that this method is efficient and reliable.

KAJIAN KEKONVERGENAN LEMAH DI RUANG HILBERT

We have the sequence , where a norm space. We have a dual space too is the collection of linier and continous functional from norm space into riil number system . If for all the sequence convergent to and then convergent to . The converse of this implication is not applicable . This study aims to explain the properties that apply to the ranks of the weak convergent and explain the relationship between the strong convergent sequence and the weak convergent sequence. From several reference sources then through the review process obtained weak nature of singularity limit sequence and relationship between the strong convergent sequence and the weak convergent sequence that if the sequence is strong convergent therefore the sequence it weak convergent.

On the dual spaces for weighted altered Cesàro and Copson spaces

We study weighted altered Cesàro and Copson spaces, which are non-ideal enlargements of the usual Cesàro and Copson spaces. Full characterizations of their dual spaces are given.

An uncountable Moore–Schmidt theorem

AbstractWe prove an extension of the Moore–Schmidt theorem on the triviality of the first cohomology class of cocycles for the action of an arbitrary discrete group on an arbitrary measure space and for cocycles with values in an arbitrary compact Hausdorff abelian group. The proof relies on a ‘conditional’ Pontryagin duality for spaces of abstract measurable maps.