On the Discrete Normal Modes of Quasigeostrophic Theory

The discrete baroclinic modes of quasigeostrophic theory are incomplete and the incompleteness manifests as a loss of information in the projection process. The incompleteness of the baroclinic modes is related to the presence of two previously unnoticed stationary step-wave solutions of the Rossby wave problem with flat boundaries. These step-waves are the limit of surface quasigeostrophic waves as boundary buoyancy gradients vanish. A complete normal mode basis for quasigeostrophic theory is obtained by considering the traditional Rossby wave problem with prescribed buoyancy gradients at the lower and upper boundaries. The presence of these boundary buoyancy gradients activates the previously inert boundary degrees of freedom. These Rossby waves have several novel properties such as the presence of multiple modes with no internal zeros, a finite number of modes with negative norms, and their vertical structures form a basis capable of representing any quasigeostrophic state with a differentiable series expansion. These properties are a consequence of the Pontryagin space setting of the Rossby wave problem in the presence of boundary buoyancy gradients (as opposed to the usual Hilbert space setting). We also examine the quasigeostrophic vertical velocity modes and derive a complete basis for such modes as well. A natural application of these modes is the development of a weakly non-linear wave-interaction theory of geostrophic turbulence that takes topography into account.

Faktor Penentu Setting Ruang Destinasi Wisata terhadap Spending Rate Wisatawan di Pantai Baron Yogyakarta

Title: Determinant Factors of Tourism Destination Space Setting to Tourist Spending Rate at Baron Beach Yogyakarta Gunungkidul Regency is popular for a beach tourist destination, committed for improving tourist’s arrival. However, for the past few years, the Original Local Government (OLG) of the tourism sector decreased by 2017, 2018, and 2019. Beach tourism provides the biggest percentage of The Original Local Government Revenue compared to other tourism destinations. One of the indicators that affects revenue from the tourism sector is the tourist spending rate. Therefore, it requires significant effort to identify the determinant factors of space settings that affect spending rates in Baron Beach. The research method used is mix method with the data collection technique used Behavioral Mapping that consist of Place Centered Mapping, Person Centered Mapping, and Time Budget. The conclusion of this research is the space setting of the Baron beach tourist destination is determined by 2 aspects of space and time. (a) The aspect of space is divided into 3 categories. They are high, medium, and low. (b) The aspect of time is divided into 3 categories. They are long, medium, and short. The determinant factors of space settings that affect spending rates in tourism destinations of Baron Beach are (a) spatial factor, (b) amenity factor (c) time factor.

Duality for Outer $$L^p_\mu (\ell ^r)$$ Spaces and Relation to Tent Spaces

We study the outer $$L^p$$ L p spaces introduced by Do and Thiele on sets endowed with a measure and an outer measure. We prove that, in the case of finite sets, for $$1< p \leqslant \infty , 1 \leqslant r < \infty $$ 1 < p ⩽ ∞ , 1 ⩽ r < ∞ or $$p=r \in \{ 1, \infty \}$$ p = r ∈ { 1 , ∞ } , the outer $$L^p_\mu (\ell ^r)$$ L μ p ( ℓ r ) quasi-norms are equivalent to norms up to multiplicative constants uniformly in the cardinality of the set. This is obtained by showing the expected duality properties between the corresponding outer $$L^p_\mu (\ell ^r)$$ L μ p ( ℓ r ) spaces uniformly in the cardinality of the set. Moreover, for $$p=1, 1 < r \leqslant \infty $$ p = 1 , 1 < r ⩽ ∞ , we exhibit a counterexample to the uniformity in the cardinality of the finite set. We also show that in the upper half space setting the desired properties hold true in the full range $$1 \leqslant p,r \leqslant \infty $$ 1 ⩽ p , r ⩽ ∞ . These results are obtained via greedy decompositions of functions in the outer $$L^p_\mu (\ell ^r)$$ L μ p ( ℓ r ) spaces. As a consequence, we establish the equivalence between the classical tent spaces $$T^p_r$$ T r p and the outer $$L^p_\mu (\ell ^r)$$ L μ p ( ℓ r ) spaces in the upper half space. Finally, we give a full classification of weak and strong type estimates for a class of embedding maps to the upper half space with a fractional scale factor for functions on $$\mathbb {R}^d$$ R d .

Right and Left Weyl Operator Matrices in a Banach Space Setting

Let X i , Y i i = 1,2 be Banach spaces. The operator matrix of the form M C = A C 0 B acting between X 1 ⊕ X 2 and Y 1 ⊕ Y 2 is investigated. By using row and column operators, equivalent conditions are obtained for M C to be left Weyl, right Weyl, and Weyl for some C ∈ ℬ X 2 , Y 1 , respectively. Based on these results, some sufficient conditions are also presented. As applications, some discussions on Hamiltonian operators are given in the context of Hilbert spaces.

Fixed point theorems of generalized multi-valued mappings in cone b-metric spaces

The aim of this paper is to establish fixed points for multivalued mappings, by adapting the ideas in [1] to the cone b-metric space setting.