scholarly journals Hardy-Type Space Associated with an Infinite-Dimensional Unitary Matrix Group

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Oleh Lopushansky
Keyword(s):  
Fock Space ◽  
Matrix Group ◽  
Unitary Matrix ◽  
Type Space ◽  
Infinite Dimensional ◽  
System A ◽  
Hardy Type ◽  
Complex Functions ◽  
Hardy Type Space ◽  
The Right

We investigate an orthogonal system of the homogenous Hilbert-Schmidt polynomials with respect to a probability measure which is invariant under the right action of an infinite-dimensional unitary matrix group. With the help of this system, a corresponding Hardy-type space of square-integrable complex functions is described. An antilinear isomorphism between the Hardy-type space and an associated symmetric Fock space is established.

scholarly journals Best Approximations in Hardy Spaces on Infinite-Dimensional Unitary Matrix Groups

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Oleh Lopushansky
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Matrix Group ◽  
Unitary Matrix ◽  
Fock Spaces ◽  
Best Approximations ◽  
Matrix Groups ◽  
Infinite Dimensional ◽  
Type Interpolation ◽  
Complex Functions

We investigate the problem of best approximations in the Hardy space of complex functions, defined on the infinite-dimensional unitary matrix group. Applying an abstract Besov-type interpolation scale and the Bernstein-Jackson inequalities, a behavior of such approximations is described. An application to best approximations in symmetric Fock spaces is shown.

𝐻𝑝 spaces for generalized Schrödinger operators and applications

2019 ◽  
Vol 31 (6) ◽  
pp. 1379-1394
Author(s):  
Yu Liu ◽  
He Wang
Keyword(s):  
Radon Measure ◽  
Riesz Transform ◽  
Auxiliary Function ◽  
Maximal Functions ◽  
Heat Semigroup ◽  
Scale Invariant ◽  
Type Space ◽  
Hardy Type ◽  
Hardy Type Space ◽  
Generalized Schrödinger Operator

AbstractIn this paper, we study the Hardy type space {H_{\mathcal{L}}^{p}({\mathbb{R}^{n}})} by means of local maximal functions associated with the heat semigroup {e^{-t\mathcal{L}}} generated by {-\mathcal{L}}, where {\mathcal{L}=-\Delta+\mu} is the generalized Schrödinger operator in {{\mathbb{R}^{n}}} ({n\geq 3}) and {\mu\not\equiv 0} is a nonnegative Radon measure satisfying certain scale-invariant Kato conditions and doubling conditions. Via the equivalence of the norms between various local maximal functions, we show that the norms {\lVert f\rVert_{H_{\mathcal{L}}^{p}({\mathbb{R}^{n}})}^{p}} and {\lVert f\rVert_{H_{m}^{p,q}({\mathbb{R}^{n}})}^{p}} are equivalent for {\frac{n}{n+\delta^{\prime}}<p\leq 1\leq q\leq\infty} ({p\neq q}) with some {\delta^{\prime}>0}. As applications, we prove that Calderón–Zygmund operators related to the auxiliary function {m(x,\mu)} are bounded from {H_{\mathcal{L}}^{p}({\mathbb{R}^{n}})} into {L^{p}({\mathbb{R}^{n}})} for {\frac{n}{n+\gamma_{1}}<p\leq 1} with {\gamma_{1}>0}. In particular, we show that the Riesz transform {\nabla(-\Delta+\mu)^{-\frac{1}{2}}}, which is a special example of the above Calderón–Zygmund operators, is bounded from {H_{\mathcal{L}}^{p}({\mathbb{R}^{n}})} into {{H}^{p}({\mathbb{R}^{n}})} for {\frac{n}{n+\gamma_{1}}<p\leq 1} with {0<\gamma_{1}<1}.

Nonholonomic deformation of coupled and supersymmetric KdV equations and Euler–Poincaré–Suslov method

2015 ◽  
Vol 27 (04) ◽  
pp. 1550011 ◽  
Author(s):  
Partha Guha
Keyword(s):  
Kdv Equation ◽  
Infinite Dimensional ◽  
Kdv Equations ◽  
Finite Dimensional ◽  
Euler Top ◽  
Two Component ◽  
Finite Dimensional Case ◽  
Coupled Kdv Equations ◽  
The Right ◽  
Supersymmetric Kdv Equation

Recently, Kupershmidt [38] presented a Lie algebraic derivation of a new sixth-order wave equation, which was proposed by Karasu-Kalkani et al. [31]. In this paper, we demonstrate that Kupershmidt's method can be interpreted as an infinite-dimensional analogue of the Euler–Poincaré–Suslov (EPS) formulation. In a finite-dimensional case, we modify Kupershmidt's deformation of the Euler top equation to obtain the standard EPS construction on SO(3). We extend Kupershmidt's infinite-dimensional construction to construct a nonholonomic deformation of a wide class of coupled KdV equations, where all these equations follow from the Euler–Poincaré–Suslov flows of the right invariant L2 metric on the semidirect product group [Formula: see text], where Diff (S1) is the group of orientation preserving diffeomorphisms on a circle. We generalize our construction to the two-component Camassa–Holm equation. We also give a derivation of a nonholonomic deformation of the N = 1 supersymmetric KdV equation, dubbed as sKdV6 equation and this method can be interpreted as an infinite-dimensional supersymmetric analogue of the Euler–Poincaré–Suslov (EPS) method.

scholarly journals Semi-device-independent framework based on natural physical assumptions

Quantum ◽  
2017 ◽  
Vol 1 ◽  
pp. 33 ◽  
Author(s):  
Thomas Van Himbeeck ◽  
Erik Woodhead ◽  
Nicolas J. Cerf ◽  
Raúl García-Patrón ◽  
Stefano Pironio
Keyword(s):  
Fock Space ◽  
Outcome Measurement ◽  
Photon Number ◽  
Quantum Correlations ◽  
Mean Value ◽  
Laser Source ◽  
Optical Scheme ◽  
Infinite Dimensional ◽  
Energy Constrained ◽  
Quantum Optical

The semi-device-independent approach provides a framework for prepare-and-measure quantum protocols using devices whose behavior must not be characterized nor trusted, except for a single assumption on the dimension of the Hilbert space characterizing the quantum carriers. Here, we propose instead to constrain the quantum carriers through a bound on the mean value of a well-chosen observable. This modified assumption is physically better motivated than a dimension bound and closer to the description of actual experiments. In particular, we consider quantum optical schemes where the source emits quantum states described in an infinite-dimensional Fock space and model our assumption as an upper bound on the average photon number in the emitted states. We characterize the set of correlations that may be exhibited in the simplest possible scenario compatible with our new framework, based on two energy-constrained state preparations and a two-outcome measurement. Interestingly, we uncover the existence of quantum correlations exceeding the set of classical correlations that can be produced by devices behaving in a purely pre-determined fashion (possibly including shared randomness). This feature suggests immediate applications to certified randomness generation. Along this line, we analyze the achievable correlations in several prepare-and-measure optical schemes with a mean photon number constraint and demonstrate that they allow for the generation of certified randomness. Our simplest optical scheme works by the on-off keying of an attenuated laser source followed by photocounting. It opens the path to more sophisticated energy-constrained semi-device-independent quantum cryptography protocols, such as quantum key distribution.

scholarly journals Endoscopic incision of protruding right ureterocele in a single collecting system: a case report

2017 ◽  
Vol 25 (4) ◽  
pp. 240-44
Author(s):  
Rinto Hariwibowo ◽  
Harrina E. Rahardjo
Keyword(s):  
External Genitalia ◽  
Excretory Urography ◽  
Voiding Cystourethrography ◽  
System A ◽  
Grade 5 ◽  
Grade Ii ◽  
The Right ◽  
Collecting System ◽  
Endoscopic Incision

Protruding ureterocele is a very rare case found in the literature. We are reporting a 21 year-old female with an intermittent protruding mass from urethra, accompanied by dysuria, hematuria, and recurrent urinary tract infection. Inspection of the external genitalia revealed a protruding mass from the urethra which could be reduced manually. Excretory urography showed bilateral single collecting systems, grade II hydronephrosis of the right kidney, and a cobra head appearance of the lower right pelvis. The patient was diagnosed with a protruding right ureterocele in a single collecting system, and thus, endoscopic incision of a ureterocele was performed. Ultrasonography which was carried out three weeks after the procedure confirmed no residual hydronephrosis or ureterocele in the bladder. Voiding cystourethrography (VCUG) underwent at a three-month-follow up revealed a grade 5 vesico-ureteral reflux (VUR) on the right side. Surgical reimplantation was then considered. In conclusion, endoscopic incision was safe and yielded good result for protruding ureteroceles, but the need for secondary surgery in several conditions should be considered.

scholarly journals White noise delta functions and continuous version theorem

1993 ◽  
Vol 129 ◽  
pp. 1-22
Author(s):  
Nobuaki Obata
Keyword(s):  
White Noise ◽  
Harmonic Analysis ◽  
Quantum Physics ◽  
Fock Space ◽  
Continuous Version ◽  
Infinite Dimensional ◽  
Schwartz Distribution ◽  
Delta Functions ◽  
Gelfand Triple ◽  
Gaussian Space

The recently developed Hida calculus of white noise [5] is an infinite dimensional analogue of Schwartz’ distribution theory besed on the Gelfand triple (E) ⊂ (L2) = L2 (E*, μ) ⊂ (E)*, where (E*, μ) is Gaussian space and (L2) is (a realization of) Fock space. It has been so far discussed aiming at an application to quantum physics, for instance [1], [3], and infinite dimensional harmonic analysis [7], [8], [13], [14], [15].

scholarly journals Double Controlled Metric Type Spaces and Some Fixed Point Results

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 320 ◽  
Author(s):  
Thabet Abdeljawad ◽  
Nabil Mlaiki ◽  
Hassen Aydi ◽  
Nizar Souayah
Keyword(s):  
Fixed Point ◽  
Triangle Inequality ◽  
Metric Spaces ◽  
Type Space ◽  
Control Functions ◽  
Right Hand ◽  
Type Spaces ◽  
The Right ◽  
The Given

In this article, in the sequel of extending b-metric spaces, we modify controlled metric type spaces via two control functions α ( x , y ) and μ ( x , y ) on the right-hand side of the b - triangle inequality, that is, d ( x , y ) ≤ α ( x , z ) d ( x , z ) + μ ( z , y ) d ( z , y ) , for all x , y , z ∈ X . Some examples of a double controlled metric type space by two incomparable functions, which is not a controlled metric type by one of the given functions, are presented. We also provide some fixed point results involving Banach type, Kannan type and ϕ -nonlinear type contractions in the setting of double controlled metric type spaces.

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