We estimate the truncated double trigonometric series ∑n=oN∑m=oMamne2πι(mx+ny),amn∈C, in Lebesgue spaces with mixed norms in terms of the pth−qth power finite double sums of its coefficients. We obtain these estimates for all possible values of the exponents involved and we provide examples of matrices in cM×N that maximize some of them up to a constant independent of M and N. We also handle the more general case where the oscillatory factors in the sum terms are not orthogonal in either variable.

Let $,T^{j,k}_{N}:L^{p}(B), ightarrow,L^{q}([0,1]),$ be the oscillatory integral operators defined by $;displaystyle T^{j,k}_{N}f(s):=int_{B} ,f(x),e^{imath N{|x|}^{j}s^{k}},dx,quad (j,k)in{1,2}^{2},,$ where $,B,$ is the unit ball in ${mathbb{R}}^{n},$ and $,N,>>1.$ We compare the asymptotic behaviour as $,N ightarrow +infty,$ of the operator norms $,parallel T^{j,k}_{N} parallel_{L^{p}(B) ightarrow L^{q}([0,1])},$ for all $,p,,qin [1,+infty].,$ We prove that, except for the dimension $n=1,,$ this asymptotic behaviour depends on the linearity or quadraticity of the phase in $s$ only. We are led to this problem by an observation on inhomogeneous Strichartz estimates for the Schr"{o}dinger equation.

In different areas of the Western Desert of Egypt, the Abu Roash “G” Member exhibits either a reservoir or source affinity. Thus, thirteen cutting samples covering the Abu Roash “G” Member were selected from the Nest-1A well at Matruh Basin to investigate its hydrocarbon source potential. Palynological age dating of the section that is calibrated with foraminifera and ostracodes enabled a proper identification of the “G” Member. Detailed analysis of the vertical distribution of particulate organic matter of this member shows two palynofacies types. PF-1 reflects an outer middle shelf depositional environment of prevailed reducing (suboxic-anoxic) conditions for the organic-rich shales of the lower “G” Member (samples 1e8). While, PF-2 reflects a minor regression that resulted in deposition of another organic-rich shales of the upper “G” Member (samples 9e13) in an inner middle shelf setting under the same prevailing reducing (suboxic-anoxic) conditions. Organic geochemical analysis reveals good to very good potential of the “G” Member as a hydrocarbon source rock (1.8e2.41, avg. 2.15 total organic content wt %). It also shows good to very good petroleum potential (PP: 4.8e11 , avg. 8 mg HC/g rock). Pyrolsis and palynofacies analyses show kerogen type II for the lower “G” Member (samples 1e8), which is characterized by high Hydrogen index (HI: 396 and 329 mg HC/g TOC at depths 1500 and 1560 m) and very high dominance of oil-prone material (amorphous organic matter “AOM”, marine palynomorphs, and sporomorphs) and very rare occurrence of gasprone material (brown phytoclasts). The upper “G” Member (samples 9e13) shows kerogen type II-III, which is characterized by a lower HI value of 213 mg HC/g TOC at depth 1340 m and it contains fewer amounts of gas-prone material and relatively lower AOM and marine palynomorphs in comparison to the upper “G” Member. Maturation parameters Tmax (430e433
C), production index (PI: 0.1 mg HC/g rock), and thermal alteration index (TAI: 2þ) indicate the lower “G” Member has already entered the early oil-window kitchen, and it is expected to produce oil. The upper “G” Member is expected to produce only oil with no gas shows, because it is marginally mature (Tmax 426
C, PI 0.2, TAI 2). The source potential index (SPI: 5.3 t HC/m2) of the “G” Member shows it as currently generating moderate quantities of oil in the area of Nest-1A well. Consequently, the organic-rich shales of the “G” Member are suggested here as a promising, active oil source rock in that extreme northwestern part of the Western Desert of Egypt. However, for commercial oil recovery from the Abu Roash “G” Member, it is highly recommended to explore the depocentre of Matruh Basin at about 150 km east the Nest-1A well.

Several kinds of synchronization of fractional-order chaotic complex systems are challenging research topics of current interest since they appear in many applications in applied sciences. Our main goal in this paper is to introduce the definition of modified projective combination-combination synchronization (MPCCS) of some fractional-order chaotic complex systems. We show that our systems are chaotic by calculating their Lyapunov exponents. The fractional Lyapunov dimension of the chaotic solutions of these systems is computed. A scheme is introduced to calculate MPCCS of four different (or identical) chaotic complex systems using the active control technique. Special cases of this type, which are projective and anti C-C synchronization, are discussed. Some figures are plotted to show that MPCCS is achieved and its errors approach zero

Several kinds of synchronization of fractional-order chaotic complex systems are challenging research topics of current interest since they appear in many applications in applied sciences. Our main goal in this paper is to introduce the definition of modified projective combination-combination synchronization (MPCCS) of some fractional-order chaotic complex systems. We show that our systems are chaotic by calculating their Lyapunov exponents. The fractional Lyapunov dimension of the chaotic solutions of these systems is computed. A scheme is introduced to calculate MPCCS of four different (or identical) chaotic complex systems using the active control technique. Special cases of this type, which are projective and anti C-C synchronization, are discussed. Some figures are plotted to show that MPCCS is achieved and its errors approach zero

Several kinds of synchronization of fractional-order chaotic complex systems are challenging research topics of current interest since they appear in many applications in applied sciences. Our main goal in this paper is to introduce the definition of modified projective combination-combination synchronization (MPCCS) of some fractional-order chaotic complex systems. We show that our systems are chaotic by calculating their Lyapunov exponents. The fractional Lyapunov dimension of the chaotic solutions of these systems is computed. A scheme is introduced to calculate MPCCS of four different (or identical) chaotic complex systems using the active control technique. Special cases of this type, which are projective and anti C-C synchronization, are discussed. Some figures are plotted to show that MPCCS is achieved and its errors approach zero

5-Chloro-3-methyl-1-phenylpyrazole-4-carbonitrile 3 was reacted with selenium in the presence of sodium borohydride and chloroacetamide to afford selanyl acetamide 5, which underwent Thorpe–Ziegler cyclization upon heating with sodium ethoxide to give the novel synthesized 4-amino- 3-methyl-1-phenyl-1H-selenolo[2,3-c]pyrazole-5-carboxamide compound (6). The latter compound was used as a versatile precursor for synthesis of other heterocyclic rings, namely pyrimidine, imidazopyrimidine and thiadiazinopyrimidine fused to selenolo[2,3-c]pyrazole moiety. The newly synthesized compounds and their derivatives were characterized by elemental and spectral analysis (IR, 1H NMR, 13C NMR and mass spectrometric analyses). Furthermore, some of these synthesized compounds were screened against various pathogenic bacterial and fungal strains. The results demonstrate that most of the synthesized compounds possess a significant antibacterial activity against gram-positive and gram-negative bacteria. Also, some of these compounds showed a remarkable antifungal activity, especially Candida albicans. On the other hand, some of the synthesized compounds possess high anti-inflammatory activity using carrageenan-induced rat paw edema assay compared with indomethacin. Graphical Abstract The present work discussed synthesis of new selenolo[2,3-c]pyrazoles fused to other heterocyclic rings, namely pyrimidine, imidazopyrimidine and thiadiazinopyrimidine. Some of the synthesized compounds showed remarkable antibacterial, antifungal and anti-inflammatory activities.

5-Chloro-3-methyl-1-phenylpyrazole-4-carbonitrile 3 was reacted with selenium in the presence of sodium borohydride and chloroacetamide to afford selanyl acetamide 5, which underwent Thorpe–Ziegler cyclization upon heating with sodium ethoxide to give the novel synthesized 4-amino- 3-methyl-1-phenyl-1H-selenolo[2,3-c]pyrazole-5-carboxamide compound (6). The latter compound was used as a versatile precursor for synthesis of other heterocyclic rings, namely pyrimidine, imidazopyrimidine and thiadiazinopyrimidine fused to selenolo[2,3-c]pyrazole moiety. The newly synthesized compounds and their derivatives were characterized by elemental and spectral analysis (IR, 1H NMR, 13C NMR and mass spectrometric analyses). Furthermore, some of these synthesized compounds were screened against various pathogenic bacterial and fungal strains. The results demonstrate that most of the synthesized compounds possess a significant antibacterial activity against gram-positive and gram-negative bacteria. Also, some of these compounds showed a remarkable antifungal activity, especially Candida albicans. On the other hand, some of the synthesized compounds possess high anti-inflammatory activity using carrageenan-induced rat paw edema assay compared with indomethacin. Graphical Abstract The present work discussed synthesis of new selenolo[2,3-c]pyrazoles fused to other heterocyclic rings, namely pyrimidine, imidazopyrimidine and thiadiazinopyrimidine. Some of the synthesized compounds showed remarkable antibacterial, antifungal and anti-inflammatory activities.

5-Chloro-3-methyl-1-phenylpyrazole-4-carbonitrile 3 was reacted with selenium in the presence of sodium borohydride and chloroacetamide to afford selanyl acetamide 5, which underwent Thorpe–Ziegler cyclization upon heating with sodium ethoxide to give the novel synthesized 4-amino- 3-methyl-1-phenyl-1H-selenolo[2,3-c]pyrazole-5-carboxamide compound (6). The latter compound was used as a versatile precursor for synthesis of other heterocyclic rings, namely pyrimidine, imidazopyrimidine and thiadiazinopyrimidine fused to selenolo[2,3-c]pyrazole moiety. The newly synthesized compounds and their derivatives were characterized by elemental and spectral analysis (IR, 1H NMR, 13C NMR and mass spectrometric analyses). Furthermore, some of these synthesized compounds were screened against various pathogenic bacterial and fungal strains. The results demonstrate that most of the synthesized compounds possess a significant antibacterial activity against gram-positive and gram-negative bacteria. Also, some of these compounds showed a remarkable antifungal activity, especially Candida albicans. On the other hand, some of the synthesized compounds possess high anti-inflammatory activity using carrageenan-induced rat paw edema assay compared with indomethacin. Graphical Abstract The present work discussed synthesis of new selenolo[2,3-c]pyrazoles fused to other heterocyclic rings, namely pyrimidine, imidazopyrimidine and thiadiazinopyrimidine. Some of the synthesized compounds showed remarkable antibacterial, antifungal and anti-inflammatory activities.

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