The Krein-von Neumann extension of a regular even order quasi-differential operator

We characterize by boundary conditions the Krein-von Neumann extension of a strictly positive minimal operator corresponding to a regular even order quasi-differential expression of Shin-Zettl type. The characterization is stated in terms of a specially chosen basis for the kernel of the maximal operator and employs a description of the Friedrichs extension due to Möller and Zettl.

Operator System Structures and Extensions of Schur Multipliers

For a given C*-algebra $\mathcal{A}$, we establish the existence of maximal and minimal operator $\mathcal{A}$-system structures on an AOU $\mathcal{A}$-space. In the case $\mathcal{A}$ is a W*-algebra, we provide an abstract characterisation of dual operator $\mathcal{A}$-systems and study the maximal and minimal dual operator $\mathcal{A}$-system structures on a dual AOU $\mathcal{A}$-space. We introduce operator-valued Schur multipliers and provide a Grothendieck-type characterisation. We study the positive extension problem for a partially defined operator-valued Schur multiplier $\varphi $ and, under some richness conditions, characterise its affirmative solution in terms of the equality between the canonical and the maximal dual operator $\mathcal{A}$-system structures on an operator system naturally associated with the domain of $\varphi $.

Principles of Electrocoagulation and Application in Wastewater Treatment

Electrocoagulation has emerged a reliable technology for the treatment of various wastewaters. Its basic principle depends on the response of water particles to strong electric field in a redox reaction. Oxidation of the anode material releases coagulating agents that form metal hydroxide complexes which neutralize particulate materials to form agglomerates. The agglomerates either settle at the bottom or float to the surface depending on the removal path of the electrocoagulation reactor. The merits of electrocoagulation include minimal sludge generation, minimal operator attention, simple equipment, high pollutant removal capacity, and ease of operation. Therefore, this chapter explores the mechanisms of electrocoagulation, components of electrocoagulation, benefits, and demerits of electrocoagulation. Furthermore, the similarity between electrocoagulation and coagulation is explored. Application of electrocoagulation for the treatment of various wastewaters was explored. Feasibility of electrocoagulation was examined through cost evaluation with other treatment technologies.

Description of the Magnetic Field and Divergence of Multisolenoid Aharonov-Bohm Potential

Explicit formulas for the magnetic field and divergence of multisolenoid Aharonov-Bohm potential are obtained; the mathematical essence of this potential is explained. It is shown that the magnetic field and divergence of this potential are very singular generalized functions concentrated at a finite number of thin solenoids. Deficiency index is found for the minimal operator generated by the Aharonov-Bohm differential expression.

Spectrums of Solvable Pantograph Differential-Operators for First Order

By using the methods of operator theory, all solvable extensions of minimal operator generated by first order pantograph-type delay differential-operator expression in the Hilbert space of vector-functions on finite interval have been considered. As a result, the exact formula for the spectrums of these extensions is presented. Applications of obtained results to the concrete models are illustrated.

Normal extensions of a first order differential operator

In the paper of W.N. Everitt and A. Zettl [26] in scalar case, all selfadjoint extensions of the minimal operator generated by Lagrange-symmetric any order quasi-differential expression with equal deficiency indexes in terms of boundary conditions are described by Glazman-Krein-Naimark method for regular and singular cases in the direct sum of corresponding Hilbert spaces of functions. In this work, by using the method of Calkin-Gorbachuk theory all normal extensions of the minimal operator generated by fixed order linear singular multipoint differential expression l = (l-, l1,... ln, l+), l-+ = d/dt + A-+, lk = d/dt + Ak where the coefficients A-+, Ak are selfadjoint operator in separable Hilbert spaces H-+, Hk, k= 1,..., n, n ? N respectively, are researched in the direct sum of Hilbert spaces of vector-functions L2(H_, (-? a))? L2(H1, (a1, b1)) ?...? L2(Hn, (an, bn)) ? L2(H+, (b,+?)) -? < a < a1 < b1 < . .. < an < bn < b < +?. Moreover, the structure of the spectrum of normal extensions is investigated. Note that in the works of A. Ashyralyev and O. Gercek [2, 3] the mixed order multipoint nonlocal boundary value problem for parabolic-elliptic equation is studied in weighed H?lder space in regular case.

Simplicity and Spectrum of Singular Hamiltonian Systems of Arbitrary Order

The paper is concerned with singular Hamiltonian systems of arbitrary order with arbitrary equal defect indices. It is proved that the minimal operator generated by the Hamiltonian system is simple. As a consequence, a sufficient condition is obtained for the continuous spectrum of every self-adjoint extension of the minimal operator to be empty in some interval and for the spectrum to be nowhere dense in this interval in terms of the numbers of linearly independent square integrable solutions.

Characterization of Eigenvalues in Spectral Gap for Singular Differential Operators

The spectral properties fornorder differential operators are considered. When given a spectral gap(a,b)of the minimal operatorT0with deficiency indexr, arbitrarympointsβi (i=1,2,…,m)in(a,b), and a positive integer functionpsuch that∑i=1mp(βi)≤r,T0has a self-adjoint extensionT̃such that eachβi (i=1,2,…,m)is an eigenvalue ofT̃with multiplicity at leastp(βi).