dual space
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2021 ◽  
Vol 127 (3) ◽  
Author(s):  
Svante Janson

We study the Banach space $D([0,1]^m)$ of functions of several variables that are (in a certain sense) right-continuous with left limits, and extend several results previously known for the standard case $m=1$. We give, for example, a description of the dual space, and we show that a bounded multilinear form always is measurable with respect to the $\sigma$-field generated by the point evaluations. These results are used to study random functions in the space. (I.e., random elements of the space.) In particular, we give results on existence of moments (in different senses) of such random functions, and we give an application to the Zolotarev distance between two such random functions.


2021 ◽  
Vol 22 (3) ◽  
pp. 365-385
Author(s):  
Honghong Zhang ◽  
Guoguo Zhang

The development of computer external storage has undergone the continuous change of perforated cassettes, tapes, floppy disks, hard disks, optical disks and flash disks. Internal memory has gone through the development of drum storage, Williams tube, mercury delay line, and magnetic core storage, until the emergence of semiconductor memory. Later RAM and ROM were born. RAM was divided into DRAM and SRAM. Due to its structure and cost advantages, DRAM has gradually developed into the widely used DDR series. At the same time, the low-power LPDDR series has also been advancing. At present, with the development of NVRAM technology, non-volatile random access memory with both internal and external storage functions is born. Dual-space storage based on NVRAM combines internal and external storage into one, and large capacity dual-space storage has become the development trend of storage.  


2021 ◽  
Vol 105 (564) ◽  
pp. 385-396
Author(s):  
Steven J. Kilner ◽  
David L. Farnsworth

We investigate the pairing of theorems about parabolas through a dual transformation. Theorems and constructions concerning a parabola in a two-dimensional space can be in one-to-one correspondence with theorems and constructions concerning a parabola in the two-dimensional dual space. These theorems are called dual theorems.


Author(s):  
Angela A. Albanese ◽  
Claudio Mele

AbstractIn this paper we continue the study of the spaces $${\mathcal O}_{M,\omega }({\mathbb R}^N)$$ O M , ω ( R N ) and $${\mathcal O}_{C,\omega }({\mathbb R}^N)$$ O C , ω ( R N ) undertaken in Albanese and Mele (J Pseudo-Differ Oper Appl, 2021). We determine new representations of such spaces and we give some structure theorems for their dual spaces. Furthermore, we show that $${\mathcal O}'_{C,\omega }({\mathbb R}^N)$$ O C , ω ′ ( R N ) is the space of convolutors of the space $${\mathcal S}_\omega ({\mathbb R}^N)$$ S ω ( R N ) of the $$\omega $$ ω -ultradifferentiable rapidly decreasing functions of Beurling type (in the sense of Braun, Meise and Taylor) and of its dual space $${\mathcal S}'_\omega ({\mathbb R}^N)$$ S ω ′ ( R N ) . We also establish that the Fourier transform is an isomorphism from $${\mathcal O}'_{C,\omega }({\mathbb R}^N)$$ O C , ω ′ ( R N ) onto $${\mathcal O}_{M,\omega }({\mathbb R}^N)$$ O M , ω ( R N ) . In particular, we prove that this isomorphism is topological when the former space is endowed with the strong operator lc-topology induced by $${\mathcal L}_b({\mathcal S}_\omega ({\mathbb R}^N))$$ L b ( S ω ( R N ) ) and the last space is endowed with its natural lc-topology.


2021 ◽  
Vol 15 (3-4) ◽  
pp. 268-272
Author(s):  
Thomas Noll

This text revisits selected aspects of Muzzulini's article and reformulates them on the basis of a three-dimensional interval space E and its dual E*. The pitch height of just intonation is conceived as an element h of the dual space. From octave-fifth-third coordinates it becomes transformed into chromatic coordinates. The dual chromatic basis is spanned by the duals a* of a minor second a and the duals b* and c*  of two kinds of augmented primes b and c. Then for every natural number n a modified pitch height form hn is derived from h by augmenting its coordinates with the factor n, followed by rounding to nearest integers. Of particular interest are the octave-consitent forms hn  mapping the octave to the value n. The three forms hn for n = 612, 118, 53 (yielding smallest deviations from the respective values of n h) form the Muzzulini basis of E*. The respective transformation matrix T* between the coordinate representations of linear forms in the Muzzulini basis and the dual chromatic basis is unimodular and a Pisot matrix with the dominant eigen-co-vector very close to h. Certain selections of the linear forms hn are displayed in Muzzuli coordinates as ball-like point clouds within a suitable cuboid containing the origin. As an open problem remains the estimation of the musical relevance of  Newton's chromatic mode, and chromatic modes in general. As a possible direction of further investigation it is proposed to study the exo-mode of Newton's chromatic mode


2021 ◽  
Vol 3 (2) ◽  
pp. 180-186
Author(s):  
Edi Kurniadi

ABSTRAKDalam artikel ini dipelajari ruang fase tereduksi dari suatu grup Lie khususnya untuk grup Lie affine  berdimensi 2. Tujuannya adalah untuk mengidentifikasi ruang fase tereduksi dari  melalui orbit coadjoint buka di titik tertentu pada ruang dual  dari aljabar Lie . Aksi dari grup Lie    pada ruang dual  menggunakan representasi coadjoint. Hasil yang diperoleh adalah ruang Fase tereduksi  tiada lain adalah orbit coadjoint-nya yang buka di ruang dual . Selanjutnya, ditunjukkan pula bahwa grup Lie affine     tepat mempunyai dua buah orbit coadjoint buka.  Hasil yang diperoleh dalam penelitian ini dapat diperluas untuk kasus grup Lie affine  berdimensi  dan untuk kasus grup Lie lainnya.ABSTRACTIn this paper, we study a reduced phase space for a Lie group, particularly for the 2-dimensional affine Lie group which is denoted by Aff (1). The work aims to identify the reduced phase space for Aff (1) by open coadjoint orbits at certain points in the dual space aff(1)* of the Lie algebra aff(1). The group action of Aff(1) on the dual space aff(1)* is considered using coadjoint representation. We obtained that the reduced phase space for the affine Lie group Aff(1) is nothing but its open coadjoint orbits. Furthermore, we show that the affine Lie group Aff (1) exactly has two open coadjoint orbits in aff(1)*. Our result can be generalized for the n(n+1) dimensional affine Lie group Aff(n) and for another Lie group.


2021 ◽  
Vol 114 ◽  
pp. 107873
Author(s):  
Ronghua Shang ◽  
Lujuan Wang ◽  
Fanhua Shang ◽  
Licheng Jiao ◽  
Yangyang Li

2021 ◽  
Vol 12 (3) ◽  
Author(s):  
Arash Ghaani Farashahi

AbstractThis paper presents a systematic study for abstract harmonic analysis on classical Banach spaces of covariant functions of characters of compact subgroups. Let G be a locally compact group and H be a compact subgroup of G. Suppose that $$\xi :H\rightarrow \mathbb {T}$$ ξ : H → T is a character, $$1\le p<\infty$$ 1 ≤ p < ∞ and $$L_\xi ^p(G,H)$$ L ξ p ( G , H ) is the set of all covariant functions of $$\xi$$ ξ in $$L^p(G)$$ L p ( G ) . It is shown that $$L^p_\xi (G,H)$$ L ξ p ( G , H ) is isometrically isomorphic to a quotient space of $$L^p(G)$$ L p ( G ) . It is also proven that $$L^q_\xi (G,H)$$ L ξ q ( G , H ) is isometrically isomorphic to the dual space $$L^p_\xi (G,H)^*$$ L ξ p ( G , H ) ∗ , where q is the conjugate exponent of p. The paper is concluded by some results for the case that G is compact.


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