Factorization of fredholm operators on analytic functions

Let Ω be a simply connected domain in the complex plane, and $\text{A(Ω}{}^{\mathrm{n}}\text{)}$, the space of functions which are defined and analytic on $\text{Ω}{}^{\mathrm{n}}$, if K is the operator on elements $\text{u(t, a}{}_1\text{, …, a}{}_{\mathrm{n}}\text{)}\text{of}\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ defined in terms of the kernels k_i(t, s, a₁, …, a_n) in $\text{A(Ω}{}^{\mathrm{n\; +\; 2}}\text{)}$ by is the identity operator on $\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$, then the operator I ? K may be factored in the form (I ? K)(M ? W) = (I ? ΠK)(M ? ΠW). Here, W is an operator on $\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ defined in terms of a kernel w(t, s, a₁, …, a_n) in $\text{A(Ω}{}^{\mathrm{n\; +\; 2}}\text{)}$ by Wu = ∝_an^tw(t, s, a₁, …, a_n) u(s, a₁, …, a_n) ds. ΠW is the operator; ΠWu = ∝_{an ? 1}w(t, s, a₁, …, a_n) u(s, a₁, …, a_n) ds. ΠK is the operator; ΠKu = ∑_{i = 1}^{n ? 1} ∝_ai^tk_i(t, s, a₁, …, a_n) ds + ∝_{an ? 1}^tk_n(t, s, a₁, …, a_n) u(s, a₁, …, a_n) ds. The operator M is of the form m(t, a₁, …, a_n)I, where $\text{m ? A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ and maps elements of $\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ into itself by multiplication. The function m is uniquely derived from K in the following manner. The operator K defines an operator $\text{K}{}^1$ on functions u in $\text{A(Ω}{}^{\mathrm{n\; +\; 2}}\text{)}$, by . A determinant $\text{δ(I ? K}{}^1\text{)}$ of the operator $\text{I ? K}{}^1$ is defined as an element $\text{m}{}^1\text{(t, a}{}_1\text{, …, a}{}_{\mathrm{n\; +\; 1}}\text{)}$ of $\text{A(Ω}{}^{\mathrm{n\; +\; 2}}\text{)}$. This is mapped into $\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ by setting a_{n + 1} = t to give m(t, a₁, …, a_n). The operator I ? ΠK may be factored in similar fashion, giving rise to a chain factorization of I ? K. In some cases all the matrix kernels k_i defining K are separable in the sense that k_i(t, s, a₁, …, a_n) = P_i(t, a₁, …, a_n) Q_i(s, a₁, …, a_n), where P_i is a 1 × p_i matrix and Q_i is a p_i × 1 matrix, each with elements in $\text{A(Ω}{}_{\mathrm{n\; +\; 1}}\text{)}$, explicit formulas are given for the kernels of the factors W. The various results are stated in a form allowing immediate extension to the vector-matrix case.     

Cyclocarbonylation of unsaturated tosylates as a method of cyclanone synthesis

A study has been made to determine the scope of the cyclocarbonylation reaction of olefinic tosylates with Na₂Fe(CO)₄.     

New measure of electron correlation

We propose to quantify the correlation inherent in a many-electron (or many-fermion) wave function psi by comparing it to the unique uncorrelated state that has the same 1-particle density operator as does /|psi>相似文献   

Forces between a single atom and its distant mirror image

An excited-state atom whose emitted light is backreflected by a distant mirror can experience trapping forces, because the presence of the mirror modifies both the electromagnetic vacuum field and the atom's own radiation reaction field. We demonstrate this mechanical action using a single trapped barium ion. We observe the trapping conditions to be notably altered when the distant mirror is translated across an optical wavelength. The well-localized barium ion enables the spatial dependence of the forces to be measured explicitly. The experiment has implications for quantum information processing and may be regarded as the most elementary optical tweezers.     

Self-sustained activity in a small-world network of excitable neurons

We study the dynamics of excitable integrate-and-fire neurons in a small-world network. At low densities p of directed random connections, a localized transient stimulus results either in self-sustained persistent activity or in a brief transient followed by failure. Averages over the quenched ensemble reveal that the probability of failure changes from 0 to 1 over a narrow range in p; this failure transition can be described analytically through an extension of an existing mean-field result. Exceedingly long transients emerge at higher densities p; their activity patterns are disordered, in contrast to the mostly periodic persistent patterns observed at low p. The times at which such patterns die out follow a stretched-exponential distribution, which depends sensitively on the propagation velocity of the excitation.     

Stability of films with nanoparticles

Numerous products (such as paints, inks, foods, and beverages) and processes (such as coating, coagulation, sedimentation, lubrication, and paper-making) depend on the stability of the liquid films separating the dispersed phase from the continuous phase in the presence of nanoparticles or surfactant micelles. Nanoparticles exhibit a tendency to form ordered layers (i.e., stratification) and particle in-layer structures in these thin films at a sufficiently high concentration, resulting in a structural force that stabilizes the dispersion. The dispersion stability depends on the film size and the nanoparticle concentration.     

Development of a laser photothermoacoustic frequency-swept system for subsurface imaging: theory and experiment

In conventional biomedical photoacoustic imaging systems, a pulsed laser is used to generate time-of-flight acoustic information of the subsurface features. This paper reports the theoretical and experimental development of a new frequency-domain (FD) photo-thermo-acoustic (PTA) principle featuring frequency sweep (chirp) and heterodyne modulation and lock-in detection of a continuous-wave laser source at 1064 nm wavelength. PTA imaging is a promising new technique which is being developed to detect tumor masses in turbid biological tissue. Owing to the linear relationship between the depth of acoustic signal generation and the delay time of signal arrival to the transducer, information specific to a particular depth can be associated with a particular frequency in the chirp signal. Scanning laser modulation with a linear frequency sweep method preserves the depth-to-delay time linearity and recovers FD-PTA signals from a range of depths. Preliminary results performed on rubber samples and solid tissue phantoms indicate that the FD-PTA technique has the potential to be a reliable tool for biomedical depth-profilometric imaging.     

Radiance-based monitoring of the extent of tissue coagulation during laser interstitial thermal therapy

Optical monitoring relates the dynamic changes in measured light intensity to the extent of treatment-induced coagulation that occurs during laser interstitial thermal therapy. We utilized a two-region Monte Carlo simulation to elucidate the nature of the changes in interstitial radiance and fluence that result from the formation of a volume of thermal coagulation surrounding a cylindrical emitter. Using simulation results, we demonstrate that radiance sensors are more sensitive than traditional fluence sensors to coagulation-induced scattering changes. Radiance measurements take advantage of directional detection angles that are more receptive to the onset and passing of the coagulation boundary. We performed experiments with albumen phantoms to demonstrate the practicality of the radiance method for monitoring interstitial laser thermal therapy.