Dual Space Research Articles

Quantifying properties (K) and (μs$\mu ^{s}$)

AbstractA Banach space X has property (K), whenever every weak* null sequence in the dual space admits a convex block subsequence so that as for every weakly null sequence in X; X has property if every weak* null sequence in admits a subsequence so that all of its subsequences are Cesàro convergent to 0 with respect to the Mackey topology. Both property and reflexivity (or even the Grothendieck property) imply property (K). In this paper, we propose natural ways for quantifying the aforementioned properties in the spirit of recent results concerning other familiar properties of Banach spaces.

On a question of Kazhdan and Yom Din

Studying a question of Kazhdan and Yom Din we first prove some fixed point theorems for linear actions of a discrete countable group G on a dual Banach space V*, and then show that the answer to the question in full generality is in the negative. In an appendix it is shown that the answer is negative also for the Banach space \({\cal B}\left( H \right)\) of bounded linear operators on an infinite-dimensional Hilbert space.

Existence of weak solutions for some quasilinear degenerated elliptic systems in weighted Sobolev spaces

We consider, for a bounded open domain Ω in Rn; (n ≥ 1) and a function u : Ω → ℝm; (m ≥ 1) the quasilinear elliptic system: (QESw)(f,g) (0.1) Which is a Dirichlet problem. Here, v belongs to the dual space , f and g satisfy some stan- dard continuity and growth conditions. we will show the existence of a weak solution of this problem in the four following cases: σ is mono- tonic, σ is strictly monotonic, σ is quasi montone and σ derives from a convex potential.

Mixed-norm Herz spaces and their applications in related Hardy spaces

In this paper, the authors introduce a class of mixed-norm Herz spaces, [Formula: see text], which is a natural generalization of mixed-norm Lebesgue spaces and some special cases of which naturally appear in the study of the summability of Fourier transforms on mixed-norm Lebesgue spaces. The authors also give their dual spaces and obtain the Riesz–Thorin interpolation theorem on [Formula: see text]. Applying these Riesz–Thorin interpolation theorem and using some ideas from the extrapolation theorem, the authors establish both the boundedness of the Hardy–Littlewood maximal operator and the Fefferman–Stein vector-valued maximal inequality on [Formula: see text]. As applications, the authors develop various real-variable theory of Hardy spaces associated with [Formula: see text] by using the existing results of Hardy spaces associated with ball quasi-Banach function spaces. These results strongly depend on the duality of [Formula: see text] and the non-trivial constructions of auxiliary functions in the Riesz–Thorin interpolation theorem.

Dual Three Phase Multilevel Space Vector Modulation Control of Diode Clamped Inverter for Dual Star Induction Motor Drive

This work suggests a dual three-phase Space Vector Modulation (SVM) for the Diode Clamped Multilevel Inverter (DCMI) to ensure a robust control of Dual Star Induction Motor (DSIM). The principal scheme is investigated to apply the same control of the six-phase multilevel inverter by two three-phase multilevel inverter to drive the DSIM. The use of classic SVM control offers significant simplifications for controlling a six phase five levels inverter. The proposed control approach employs according to the hybridization of various conversion functions to establish the modulation strategy for each voltage vector and its placement in the plane of voltage modulation in distinctly and easy manner. A numerical simulation under MATLAB/Simulink is carried out to evaluate the Indirect Field Oriented Control (IFOC) of DSIM drive fed by multilevel inverters. The simulation outcomes clearly reveal good performance of the designed control strategy in terms of THD and control efficiency.

Tsallis value-at-risk: generalized entropic value-at-risk

AbstractMotivated by Ahmadi-Javid (Journal of Optimization Theory Applications, 155(3), 2012, 1105–1123) and Ahmadi-Javid and Pichler (Mathematics and Financial Economics, 11, 2017, 527–550), the concept of Tsallis Value-at-Risk (TsVaR) based on Tsallis entropy is introduced in this paper. TsVaR corresponds to the tightest possible upper bound obtained from the Chernoff inequality for the Value-at-Risk. The main properties and analogous dual representation of TsVaR are investigated. These results partially generalize the Entropic Value-at-Risk by involving Tsallis entropies. Three spaces, called the primal, dual, and bidual Tsallis spaces, corresponding to TsVaR are fully studied. It is shown that these spaces equipped with the norm induced by TsVaR are Banach spaces. The Tsallis spaces are related to the $L^p$ spaces, as well as specific Orlicz hearts and Orlicz spaces. Finally, we derive explicit formula for the dual TsVaR norm.

Finite Element Approximation of the Levi-Civita Connection and Its Curvature in Two Dimensions

We construct finite element approximations of the Levi-Civita connection and its curvature on triangulations of oriented two-dimensional manifolds. Our construction relies on the Regge finite elements, which are piecewise polynomial symmetric (0, 2)-tensor fields possessing single-valued tangential-tangential components along element interfaces. When used to discretize the Riemannian metric tensor, these piecewise polynomial tensor fields do not possess enough regularity to define connections and curvature in the classical sense, but we show how to make sense of these quantities in a distributional sense. We then show that these distributional quantities converge in certain dual Sobolev norms to their smooth counterparts under refinement of the triangulation. We also discuss projections of the distributional curvature and distributional connection onto piecewise polynomial finite element spaces. We show that the relevant projection operators commute with certain linearized differential operators, yielding a commutative diagram of differential complexes.

Inequivalent representations of the dual space

Inequivalent representations of the dual space

Numerical ranges and complex symmetric operators in semi-inner-product spaces

We introduce the numerical range of a bounded linear operator on a semi-inner-product space. We compute the numerical ranges of some operators on ell _{2}^{p}(mathbb{C})(1le p < infty ) and show that the numerical range of the backward shift on an infinite-dimensional space ell ^{p} is the open unit disc. We define a conjugation and a complex symmetric operator on a semi-inner-product space and discuss complex symmetry in the dual space. We prove some properties of a generalized adjoint of a complex symmetric operator. We also show that the numerical range of the complex conjugation on ell _{n}^{p}(n ge 2) is the closed unit disc. Finally, we discuss the sequentially essential numerical ranges of operators on a semi-inner-product space.

On the Functorial Properties of the p-Analog of the Fourier–Stieltjes Algebras

In this paper, some known results about the functorial properties of the Fourier–Stieltjes algebra, B(G), will be generalized. First of all, the idempotent theorem on the Fourier–Stieltjes algebra will be promoted and linked to the p-analog one. Next, the p-analog of the \(\pi\)-Fourier space introduced by Arsac will be given, and by taking advantage of the theory of ultrafilters, the connection between the dual space of the algebra of p-pseudofunctions and the p-analog of the \(\pi\)-Fourier space will be fully investigated. As the main result, one of the significant and applicable functorial properties of the p-analog of the Fourier–Stieltjes algebras will be achieved.

On the reflexivity of the spaces of variable integrability and summability

In this paper, we show that under the condition \( 1<p_-, q_-, p_+, q_+<\infty \), the space \(\ell ^{q(\cdot )} (L^{p(\cdot )})\) is reflexive as long as \(\ell ^{q(\cdot )} (L^{p(\cdot )})\) is a Banach space. In this way we give an answer to the open problem posed by Hästö in 2017 about the reflexivity of the variable mixed Lebesgue-sequence spaces \(\ell ^{q(\cdot )} (L^{p(\cdot )})\). What is important here is that the dual space of \(\ell ^{q(\cdot )} (L^{p(\cdot )})\) is specified. As its direct corollary, we show that the corresponding Besov space \(B^{s(\cdot )}_{p(\cdot )q(\cdot )}\) is reflexive.

Hermitian K-theory for stable infty -categories I: Foundations

This paper is the first in a series in which we offer a new framework for hermitian {text {K}}-theory in the realm of stable infty -categories. Our perspective yields solutions to a variety of classical problems involving Grothendieck-Witt groups of rings and clarifies the behaviour of these invariants when 2 is not invertible. In the present article we lay the foundations of our approach by considering Lurie’s notion of a Poincaré infty -category, which permits an abstract counterpart of unimodular forms called Poincaré objects. We analyse the special cases of hyperbolic and metabolic Poincaré objects, and establish a version of Ranicki’s algebraic Thom construction. For derived infty -categories of rings, we classify all Poincaré structures and study in detail the process of deriving them from classical input, thereby locating the usual setting of forms over rings within our framework. We also develop the example of visible Poincaré structures on infty -categories of parametrised spectra, recovering the visible signature of a Poincaré duality space. We conduct a thorough investigation of the global structural properties of Poincaré infty -categories, showing in particular that they form a bicomplete, closed symmetric monoidal infty -category. We also study the process of tensoring and cotensoring a Poincaré infty -category over a finite simplicial complex, a construction featuring prominently in the definition of the {text {L}}- and Grothendieck-Witt spectra that we consider in the next instalment. Finally, we define already here the 0th Grothendieck-Witt group of a Poincaré infty -category using generators and relations. We extract its basic properties, relating it in particular to the 0th {text {L}}- and algebraic {text {K}}-groups, a relation upgraded in the second instalment to a fibre sequence of spectra which plays a key role in our applications.

Some estimates for Riesz transforms associated with Schrödinger operators

Let L = −Δ+V be the Schr¨odinger operator on Rn, where n ≥ 3, and nonnegative potential V belongs to the reverse H¨older class RHq with n/2 ≤ q &lt; n. Let Hp L(Rn) denote the Hardy space related to L and BMOL(Rn) denote the dual space of H1L (Rn). In this paper, we show that Tα,β = V α∇L−β is bounded from Hp1 L (Rn) into Lp2 (Rn) for n n+δ′ &lt; p1 ≤ 1 and 1 p2 = 1 p1 − 2(β−α) n , where δ′ = min{1, 2−n/q0}, and q0 is the reverse H¨older index of V. Moreover, we prove T∗ α,β is bounded on BMOL(Rn) when β − α = 1/2.

Recurrence properties for linear dynamical systems: An approach via invariant measures

We study different pointwise recurrence notions for linear dynamical systems from the Ergodic Theory point of view. We show that from any reiteratively recurrent vector x0, for an adjoint operator T on a separable dual Banach space X, one can construct a T-invariant probability measure which contains x0 in its support. This allows us to establish some equivalences, for these operators, between some strong pointwise recurrence notions which in general are completely distinguished. In particular, we show that (in our framework) reiterative recurrence coincides with frequent recurrence; for complex Hilbert spaces uniform recurrence coincides with the property of having a spanning family of unimodular eigenvectors; and the same happens for power-bounded operators on complex reflexive Banach spaces. These (surprising) properties are easily generalized to product and inverse dynamical systems, which implies some relations with the respective hypercyclicity notions. Finally we study how typical is an operator with a non-zero reiteratively recurrent vector in the sense of Baire category.

Compact retractions and Schauder decompositions in Banach spaces

Let X X be a separable Banach space. We give an almost characterization of the existence of a Finite Dimensional Decomposition (FDD for short) for X X in terms of Lipschitz retractions onto generating compact subsets K K of X X . In one direction, if X X admits an FDD then we construct a Lipschitz retraction onto a small generating convex and compact set K K . On the other hand, we prove that if X X admits a “small” generating compact Lipschitz retract then X X has the π \pi -property. It is still unknown if the π \pi -property is isomorphically equivalent to the existence of an FDD. For dual Banach spaces this is true, so our results give a characterization of the FDD property for dual Banach spaces X X . We give an example of a small generating convex compact set which is not a Lipschitz retract of C [ 0 , 1 ] C[0,1] , although it is contained in a small convex Lipschitz retract and contains another one. We characterize isomorphically Hilbertian spaces as those Banach spaces X X for which every convex and compact subset is a Lipschitz retract of X X . Finally, we prove that a convex and compact set K K in any Banach space with a Uniformly Rotund in Every Direction norm is a uniform retract, of every bounded set containing it, via the nearest point map.

Convolution in dual Cesàro sequence spaces

We investigate convolution operators in the sequence spaces dp, for 1≤p<∞. These spaces, for p>1, arise as dual spaces of the Cesàro sequence spaces cesp thoroughly investigated by G. Bennett. A detailed study is also made of the algebra of those sequences which convolve dp into dp. It turns out that such multiplier spaces exhibit features which are very different to the classical multiplier spaces of ℓp.

Time regularity of stochastic convolutions and stochastic evolution equations in duals of nuclear spaces

Let be a locally convex space and let be a quasi-complete bornological nuclear space (like spaces of smooth functions and distributions) with dual spaces and In this work we introduce sufficient conditions for time regularity properties of the -valued stochastic convolution where is the dual semigroup to a C 0-semigroup on is a suitable operator form into and M is a cylindrical-martingale valued measure on Our result is latter applied to study time regularity of solutions to -valued stochastic evolutions equations.

Towards explicit discrete holography: Aperiodic spin chains from hyperbolic tilings

We propose a new example of discrete holography that provides a new step towards establishing the AdS/CFT duality for discrete spaces. A class of boundary Hamiltonians is obtained in a natural way from regular tilings of the hyperbolic Poincaré disk, via an inflation rule that allows to construct the tiling using concentric layers of tiles. The models in this class are aperiodic spin chains, whose sequences of couplings are obtained from the bulk inflation rule. We explicitly choose the aperiodic XXZ spin chain with spin 1/2 degrees of freedom as an example. The properties of this model are studied by using strong disorder renormalization group techniques, which provide a tensor network construction for the ground state of this spin chain. This can be regarded as discrete bulk reconstruction. Moreover we compute the entanglement entropy in this setup in two different ways: a discretization of the Ryu-Takayanagi formula and a generalization of the standard computation for the boundary aperiodic Hamiltonian. For both approaches, a logarithmic growth of the entanglement entropy in the subsystem size is identified. The coefficients, i.e. the effective central charges, depend on the bulk discretization parameters in both cases, albeit in a different way.

On Dual B-Topological Spaces Determined by Filterbase and Some Sets in a Dual B-algebra

This paper presents dual B-topologies that are determined by filterbase and some sets in a dual B-algebra. Also, some properties of a filterbase in a dual B-topological space are provided. In particular, a commutative dual B-topological space and a symmetric B-topological space are topological dual B-algebras.

The Application of Euler-Rodrigues Formula over Hyper-Dual Matrices

The Lie group over the hyper-dual matrices and its corresponding Lie algebra are first introduced in this study. One of Euler's strategies called the Euler-Rodrigues formula is applied to the matrices of hyper-dual rotations. The fundamental relationship between the hyper-dual numbers and the dual numbers allows us to apply the formula on dual lines and two intersecting real lines in the three-dimensional dual and Euclidean spaces, respectively.