SPECTRUM DISTRIBUTION OF THE SEDOND ORDER GENERALIZED DISTRIBUTED PARAMETER SYSTEMS

Spectrum distribution of the second order generalized distributed parameter system was discussed via the functional analysis and operator theory in Hilbert space. The solutions of the problem and the constructive expression of the solutions are given by the generalized inverse one of bounded linear operator. This is theoretically important for studying the stabilization and asymptotic stability of the second order generalized distributed parameter system.

Generalized Parameters of Porous Materials as Similarity Numbers

In the multifactorial preparation of porous materials, the simultaneous/se- quential influence of a number of technological variables changes the individual parameters of the texture of the material (surface area, volume, pore size, etc.) to different values and with increase or decrease. Generalized parameters (GPs) combine these changes;new dependencies arise. GPs behave like the dimensionless similarity numbers known in science and technology (Reynolds, etc.). They split the data (phenomena) into series with similar properties, reveal special patterns and structural nuances. New GPs proposed. The average pore size is presented as the product of two GPs: the dimentionless shape factor F and pore width of unknown shape (reciprocal of the volumetric surface). Using F, for example, the SBA-15 dataset (D. Zhao, Science 1998) was split into 3 series of samples differing in synthesis temperatures, unit cell parameters, intra-wall pore volumes, pore lengths, and the ratios of wall thickness to pore size. A surprising phenomenon was discovered one of the copolymers acts in a similar way to high temperatures. The standard deviation (STD, %) of the texture parameter in the series is its serial GP. The surface topography (micropore volume per m2) is proposed;it eliminates fluctuation in material density and has a lower STD than cm3/g. Examples of the use of GPs for silica, carbon, alumina and catalysts are given. A correlation has been shown between the efficiency of some catalytic reactions (adsorption) and GPs. GPs provide new information about materials and open up new research challenges.

Adsorbents for Gas Storage: Gas Energy, Sorbent Energy and Their Relationship to Capacity

Generalized variables make it possible to reveal the nuances of the structure of porous materials and divide samples into their series with similar properties (Titelman, L. AMPC 2021, vol. 11, No. 11). Adsorbents for gas storage have a unique set of variables that can be combined: textural and mechanical properties of the adsorbent, preparation conditions, pressure and temperature of gas during storage and delivery. Taking gas pressure and mechanical strength as forces, textural properties as displacements, we obtained the energies of gas and sorbent as generalized variables. The interrelationships between them and the storage capacity for metal-organic frameworks, porous organic polymers and activated carbons were studied. Due to the variety of sorbents and the attracting effect of micropore walls on gas adsorption, the previously proposed average thickness of the probing gas layer is useful as estimation of the pore size. Its effect on adsorbent capacity was tested. The ratio of the gas layer to the kinetic diameter of the molecule gives the packing of molecules inside the pores and makes it possible to represent the pore model. Excessive surface area results in too small pores, repulsive forces and reduced capacitance. Sometimes the gas energy correlates better with the residual adsorption uptake than with the total or delivery capacity. Compared to texture parameters, the proposed generalized variables correlate better with sorbent capacity.

Generalized Intersection Bodies with Parameter

In 2005, the classical intersection bodies and L_p intersection bodies were extended. Afterwards, the concept of gen-eral L_p intersection bodies and the generalized intersection bodies was introduced. In this paper, we define the generalized bodies with parameter. Besides, we establish the extremal values for volume, Brunn-Minkowski type inequality for radial combination and L_p harmonic Blaschke combination of this notion.