In this paper,we mainly use the Frechet derivative to extend the Bohr inequality with a lacunary series to the higher-dimensional space,namely,mappings from Unto U(resp.Unto Un).In addition,we discuss whether or not t...In this paper,we mainly use the Frechet derivative to extend the Bohr inequality with a lacunary series to the higher-dimensional space,namely,mappings from Unto U(resp.Unto Un).In addition,we discuss whether or not there is a constant term in f,and we obtain two redefined Bohr inequalities in Un.Finally,we redefine the Bohr inequality of the odd and even terms of the analytic function f so as to obtain conclusions for two different higher-dimensional alternating series.展开更多
The development of polyoxometalates for olefin oxidation is critical to achieving the green chemical process of the C5 fraction further processing.Di-lacunary silicotungstic anions were easily obtained by continuously...The development of polyoxometalates for olefin oxidation is critical to achieving the green chemical process of the C5 fraction further processing.Di-lacunary silicotungstic anions were easily obtained by continuously adjusting the p H instead of the traditional step-by-step method,which exhibited excellent performance in the catalytic oxidation of cyclopentene(CPE)to aldehydes or alcohols.The 93.69%CPE conversion and 97.15%total product selectivity(41.38%for glutaraldehyde(GA)and 55.77%for 1,2-cyclopentanediol(1,2-diol)were achieved by using H_(2)O_(2)as the oxidant and acetonitrile as the solvent.Through complementary characterization,it was found that the optimized di-lacunary silicotungstic polyoxometalate retained a complete Keggin structure,and exhibited better catalytic activity and stability than the mono-lacunary or saturated silicodecatungstate because it exposed more catalytic active centers.Furthermore,in situ FT-IR spectra was utilized to monitor the reaction process,revealing the formation of the active species W(O_(2))on the di-lacunary silicotungstic polyoxometalate and the intermediate epoxycyclopentane during the catalytic oxidation of cyclopentene.展开更多
In this paper, we define a new generalized difference sequence space involving lacunary sequence. Then, we examine k-NUC property and property （β） for this space and also show that it is not rotund where p = （pr）...In this paper, we define a new generalized difference sequence space involving lacunary sequence. Then, we examine k-NUC property and property （β） for this space and also show that it is not rotund where p = （pr） is a bounded sequence of positive real numbers with pr ≥ 1 for all r ∈ N.展开更多
In this article, we introduce the concept of lacunary statistical convergence of order a of real number sequences and give some inclusion relations between the sets of lacu- nary statistical convergence of order α an...In this article, we introduce the concept of lacunary statistical convergence of order a of real number sequences and give some inclusion relations between the sets of lacu- nary statistical convergence of order α and strong Nα （p）-summability. Furthermore, some relations between the spaces Nθα （p） and Sθα are examined.展开更多
he present paper covers tlie preparation and isolation of a series of monosub-stituted molybdophosphates Na_5 [PZ(H_2O)Mo_11 O_39]· 5H_2O (Z=Mn￣2+, Co￣2+,Ni￣2+, Cu￣2+, Zn￣2+)by an improved method. These comp...he present paper covers tlie preparation and isolation of a series of monosub-stituted molybdophosphates Na_5 [PZ(H_2O)Mo_11 O_39]· 5H_2O (Z=Mn￣2+, Co￣2+,Ni￣2+, Cu￣2+, Zn￣2+)by an improved method. These complexes were characterized byfull elemental analysis, IR spectra and UV-Vis. The studies show that the pre-pared products still keep Keggin structure and the substituting metal would belooated in coordination site. Based on this, their some pysico-cliemical propertiesrelated with their catalytic beliaviour,such as thermal stability and acidity proper-ty, were studied.展开更多
The purpose of this paper is to introduce and study some sequence spaces which are defined by combining the concepts of sequences of Musielak-Orlicz functions, invariant means and lacunary convergence on 2-norm space....The purpose of this paper is to introduce and study some sequence spaces which are defined by combining the concepts of sequences of Musielak-Orlicz functions, invariant means and lacunary convergence on 2-norm space. We establish some inclusion relations between these spaces under some conditions.展开更多
In this study, we define (f, p)-Asymptotically Lacunary Equivalent Sequences with respect to the ideal I using a non-trivial ideal , a lacunary sequence , a strictly positive sequence , and a modulus function f, and o...In this study, we define (f, p)-Asymptotically Lacunary Equivalent Sequences with respect to the ideal I using a non-trivial ideal , a lacunary sequence , a strictly positive sequence , and a modulus function f, and obtain some revelent connections between these notions.展开更多
The saturation rate and class of (0,m1,m2, …,mq) trigonometric inter polation operators in . spaces have been determined by Cavaretta and Selvaraj. In this paper, we consider the convergence and saturation problems o...The saturation rate and class of (0,m1,m2, …,mq) trigonometric inter polation operators in . spaces have been determined by Cavaretta and Selvaraj. In this paper, we consider the convergence and saturation problems of these operators in (1≤p≤∞) and obtain complete results.展开更多
In this paper, we introduce and study the concept of lacunary invariant convergence for sequences of sets with respect to modulus functionfand give some inclusion relations.
Synthesis of uranyl polyoxometalate clusters attract more attention but remain great challenging.Herein,two uranyl polyoxometalate inorganic clusters of{Na_(6)(NH_(4))_(19)[(TeW_(9)O_(33))_(4)[TeO_(2)(OH)](UO_(2))4(H_...Synthesis of uranyl polyoxometalate clusters attract more attention but remain great challenging.Herein,two uranyl polyoxometalate inorganic clusters of{Na_(6)(NH_(4))_(19)[(TeW_(9)O_(33))_(4)[TeO_(2)(OH)](UO_(2))4(H_(2)O)_(2)]·50H_(2)O}n(1,Te-U_(4))and(NH_(4))18{Na@[(TeW_(9)O_(32)OH)(TeW_(9)O_(33))_(2)[TeO(OH)][W_(3)O_(12)(μ_(3)-OH)](UO_(2))_(6)(μ_(2)-OH)_(2)H_(2)O]}·75H_(2)O(2,Te-U6)were prepared using trilacunary[α-B-TeW_(9)O_(33)]8-as inorganic ligands.Structural analysis showed that compound 1 is a one-dimensional chain structure formed by tetrameric aggregates of[(α-B-TeW_(9)O_(33))_(4)(TeO_(2)(OH))(UO_(2))_(4)(H_(2)O)2]25-as nodes and Na+ions as linkers.The anionic tetrameric aggregates in 1 is made of two sandwich-type dimers of[(TeW_(9)O_(33))_(2)(UO_(2))_(2)H_(2)O]12-bridged by a[TeO_(2)(OH)]-unit.Compound 2 is a triangular trimeric architecture templated by Na+ion.The solid-state luminescence emission spectra of compounds 1 and 2 show the characteristic peaks of uranyl ions.Compound 1 with one-dimensional chain structure exhibited a higher photoluminescence quantum yields(PLQY)than that of 2,due to the more rigidity of 1 reducing the nonradiative transitions.展开更多
In this article we introduce the generalized lacunary difference sequence spaces [N_θ,M,Δ~m]_0,[N_θ,M,Δ~m]_1 and [N_θ,M,Δ~m]_∞ using m^(th)- difference.We study their properties like completeness,solidness,symm...In this article we introduce the generalized lacunary difference sequence spaces [N_θ,M,Δ~m]_0,[N_θ,M,Δ~m]_1 and [N_θ,M,Δ~m]_∞ using m^(th)- difference.We study their properties like completeness,solidness,symmetricity.Also we obtain some inclusion relations involving the spaces [N_θ,M,Δ~m]_0,[N_θ,M,Δ~m]_1 and[N_θ,M,Δ~m]_∞ and the Cesàro summable and strongly Cesàro summable sequences.展开更多
Olefins and allylic alcohols have been epoxidized with commercially available hydrogen peroxide (30% H2O2) using a phase transfer catalyst,composed of cetyltrimethylammonium cations and a lacunary-type phosphotungstat...Olefins and allylic alcohols have been epoxidized with commercially available hydrogen peroxide (30% H2O2) using a phase transfer catalyst,composed of cetyltrimethylammonium cations and a lacunary-type phosphotungstate anion [PW11O39]7-or the complete Keggin-type heteropolyanion [PW12O40]3-,under two-phase conditions using ethyl acetate as the solvent. It was found that the lacunary-type catalyst showed higher activity and better recyclability than the complete Keggin-type catalyst under the same reaction conditions. 31P NMR spectroscopy and solubility measurements for the two catalysts revealed that the [PW11O39]7-anion had a much faster degradation rate than the [PW12O40]3-anion in an excess of H2O2,which resulted in the formation of more catalytically active species. As a result,the lacunary-type phosphotungstate anion-based catalyst gave a better catalytic performance than the complete Keggin-type anion in ethyl acetate.展开更多
For a prescribed set of lacunary data{(x_(ν),f_(ν),f^(″)_(ν)):ν=0,1,...,N}with equally spaced knot sequence in the unit interval,we show the existence of a fam-ily of fractal splines S^(α)_(b)∈C 3[0,1]satisfyin...For a prescribed set of lacunary data{(x_(ν),f_(ν),f^(″)_(ν)):ν=0,1,...,N}with equally spaced knot sequence in the unit interval,we show the existence of a fam-ily of fractal splines S^(α)_(b)∈C 3[0,1]satisfying S^(α)_(b)(x_(ν))=f_(ν),(S^(α)_(b))^(2)(x_(ν))=f^(″)_(ν)forν=0,1,...,N and suitable boundary conditions.To this end,the unique quintic spline introduced by A.Meir and A.Sharma[SIAM J.Numer.Anal.10(3)1973,pp.433-442]is generalized by using fractal functions with variable scaling pa-rameters.The presence of scaling parameters that add extra“degrees of freedom”,self-referentiality of the interpolant,and“fractality”of the third derivative of the in-terpolant are additional features in the fractal version,which may be advantageous in applications.If the lacunary data is generated from a functionΦsatisfying certain smoothness condition,then for suitable choices of scaling factors,the corresponding fractal spline S^(α)_(b)satisfies||Φ^(r)−(S^(α)_(b))(r)||∞→0 for 0≤r≤3,as the number of partition points increases.展开更多
In this paper, we investigate the growth and value distribution of meromorphic solutions to higher order linear differential equations with some dominating coefficient being Lacunary series and the results of this pap...In this paper, we investigate the growth and value distribution of meromorphic solutions to higher order linear differential equations with some dominating coefficient being Lacunary series and the results of this paper improve and extend the previous results of J. Tu, 2013.展开更多
A new heteropolycomplex, K 6H 3[ZnW 11 O 40 Al]·9 5H 2O was prepared and characterized by means of elemental analysis, IR, UV spectroscopy, 27 Al NMR , electrochemistry and X ray crystall...A new heteropolycomplex, K 6H 3[ZnW 11 O 40 Al]·9 5H 2O was prepared and characterized by means of elemental analysis, IR, UV spectroscopy, 27 Al NMR , electrochemistry and X ray crystallography. The crystal of K 6H 3[ZnW 11 O 40 Al]·9 5H 2O is cubic, space group Fm 3m , with lattice constants a=b=c =2 144 8(2) nm, V =9 866(2) nm 3, Mo radiation, R =0 057 8 for 497 independent data with [ I>2δ(I) ]. The anion is of α type Keggin structure with C s symmetry.展开更多
In this paper, we have studied the Lacunary type of polynomials and proved a result which generalizes as well as refines some well-known polynomial inequalities regarding the growth of polynomials not vanishing inside...In this paper, we have studied the Lacunary type of polynomials and proved a result which generalizes as well as refines some well-known polynomial inequalities regarding the growth of polynomials not vanishing inside a circle. Further the paper corrects the proofs of some already known results.展开更多
Polyoxometalates(POMs)have been emerging as one of the most widely-studied interdisciplinary research fields of inorganic chemistry due to their structural diversities and versatile physicochemical properties.The inco...Polyoxometalates(POMs)have been emerging as one of the most widely-studied interdisciplinary research fields of inorganic chemistry due to their structural diversities and versatile physicochemical properties.The incorporation of transition metals into lacunary POM ligands can lead to the formation of various transition-metal-substituted POMs(TMSPs)with wide potential applications in research areas including medicine,magnetism,functional materials,catalysis,and energy storage.In this review,we summarized the recent advances of mixed-transition-metal-substituted POMs(Mixed TMSPs)with respect to their synthetic strategies,structural diversities,and relevant applications.Finally,a comprehensive outlook of this research area was also prospected.展开更多
基金supported by Guangdong Natural Science Foundations(2021A1515010058)。
文摘In this paper,we mainly use the Frechet derivative to extend the Bohr inequality with a lacunary series to the higher-dimensional space,namely,mappings from Unto U(resp.Unto Un).In addition,we discuss whether or not there is a constant term in f,and we obtain two redefined Bohr inequalities in Un.Finally,we redefine the Bohr inequality of the odd and even terms of the analytic function f so as to obtain conclusions for two different higher-dimensional alternating series.
基金the Science and Technology Project of Maoming(China)(200203094555139)for financial support。
文摘The development of polyoxometalates for olefin oxidation is critical to achieving the green chemical process of the C5 fraction further processing.Di-lacunary silicotungstic anions were easily obtained by continuously adjusting the p H instead of the traditional step-by-step method,which exhibited excellent performance in the catalytic oxidation of cyclopentene(CPE)to aldehydes or alcohols.The 93.69%CPE conversion and 97.15%total product selectivity(41.38%for glutaraldehyde(GA)and 55.77%for 1,2-cyclopentanediol(1,2-diol)were achieved by using H_(2)O_(2)as the oxidant and acetonitrile as the solvent.Through complementary characterization,it was found that the optimized di-lacunary silicotungstic polyoxometalate retained a complete Keggin structure,and exhibited better catalytic activity and stability than the mono-lacunary or saturated silicodecatungstate because it exposed more catalytic active centers.Furthermore,in situ FT-IR spectra was utilized to monitor the reaction process,revealing the formation of the active species W(O_(2))on the di-lacunary silicotungstic polyoxometalate and the intermediate epoxycyclopentane during the catalytic oxidation of cyclopentene.
文摘In this paper, we define a new generalized difference sequence space involving lacunary sequence. Then, we examine k-NUC property and property （β） for this space and also show that it is not rotund where p = （pr） is a bounded sequence of positive real numbers with pr ≥ 1 for all r ∈ N.
文摘In this article, we introduce the concept of lacunary statistical convergence of order a of real number sequences and give some inclusion relations between the sets of lacu- nary statistical convergence of order α and strong Nα （p）-summability. Furthermore, some relations between the spaces Nθα （p） and Sθα are examined.
文摘he present paper covers tlie preparation and isolation of a series of monosub-stituted molybdophosphates Na_5 [PZ(H_2O)Mo_11 O_39]· 5H_2O (Z=Mn￣2+, Co￣2+,Ni￣2+, Cu￣2+, Zn￣2+)by an improved method. These complexes were characterized byfull elemental analysis, IR spectra and UV-Vis. The studies show that the pre-pared products still keep Keggin structure and the substituting metal would belooated in coordination site. Based on this, their some pysico-cliemical propertiesrelated with their catalytic beliaviour,such as thermal stability and acidity proper-ty, were studied.
文摘The purpose of this paper is to introduce and study some sequence spaces which are defined by combining the concepts of sequences of Musielak-Orlicz functions, invariant means and lacunary convergence on 2-norm space. We establish some inclusion relations between these spaces under some conditions.
文摘In this study, we define (f, p)-Asymptotically Lacunary Equivalent Sequences with respect to the ideal I using a non-trivial ideal , a lacunary sequence , a strictly positive sequence , and a modulus function f, and obtain some revelent connections between these notions.
文摘The saturation rate and class of (0,m1,m2, …,mq) trigonometric inter polation operators in . spaces have been determined by Cavaretta and Selvaraj. In this paper, we consider the convergence and saturation problems of these operators in (1≤p≤∞) and obtain complete results.
文摘In this paper, we introduce and study the concept of lacunary invariant convergence for sequences of sets with respect to modulus functionfand give some inclusion relations.
基金the National Natural Science Foundation of China(Grant Nos.21901002,92161104,21871224,92161203,and 21721001)。
文摘Synthesis of uranyl polyoxometalate clusters attract more attention but remain great challenging.Herein,two uranyl polyoxometalate inorganic clusters of{Na_(6)(NH_(4))_(19)[(TeW_(9)O_(33))_(4)[TeO_(2)(OH)](UO_(2))4(H_(2)O)_(2)]·50H_(2)O}n(1,Te-U_(4))and(NH_(4))18{Na@[(TeW_(9)O_(32)OH)(TeW_(9)O_(33))_(2)[TeO(OH)][W_(3)O_(12)(μ_(3)-OH)](UO_(2))_(6)(μ_(2)-OH)_(2)H_(2)O]}·75H_(2)O(2,Te-U6)were prepared using trilacunary[α-B-TeW_(9)O_(33)]8-as inorganic ligands.Structural analysis showed that compound 1 is a one-dimensional chain structure formed by tetrameric aggregates of[(α-B-TeW_(9)O_(33))_(4)(TeO_(2)(OH))(UO_(2))_(4)(H_(2)O)2]25-as nodes and Na+ions as linkers.The anionic tetrameric aggregates in 1 is made of two sandwich-type dimers of[(TeW_(9)O_(33))_(2)(UO_(2))_(2)H_(2)O]12-bridged by a[TeO_(2)(OH)]-unit.Compound 2 is a triangular trimeric architecture templated by Na+ion.The solid-state luminescence emission spectra of compounds 1 and 2 show the characteristic peaks of uranyl ions.Compound 1 with one-dimensional chain structure exhibited a higher photoluminescence quantum yields(PLQY)than that of 2,due to the more rigidity of 1 reducing the nonradiative transitions.
文摘In this article we introduce the generalized lacunary difference sequence spaces [N_θ,M,Δ~m]_0,[N_θ,M,Δ~m]_1 and [N_θ,M,Δ~m]_∞ using m^(th)- difference.We study their properties like completeness,solidness,symmetricity.Also we obtain some inclusion relations involving the spaces [N_θ,M,Δ~m]_0,[N_θ,M,Δ~m]_1 and[N_θ,M,Δ~m]_∞ and the Cesàro summable and strongly Cesàro summable sequences.
基金supported by the National Natural Science Foundation of China (20773037, 21073058)
文摘Olefins and allylic alcohols have been epoxidized with commercially available hydrogen peroxide (30% H2O2) using a phase transfer catalyst,composed of cetyltrimethylammonium cations and a lacunary-type phosphotungstate anion [PW11O39]7-or the complete Keggin-type heteropolyanion [PW12O40]3-,under two-phase conditions using ethyl acetate as the solvent. It was found that the lacunary-type catalyst showed higher activity and better recyclability than the complete Keggin-type catalyst under the same reaction conditions. 31P NMR spectroscopy and solubility measurements for the two catalysts revealed that the [PW11O39]7-anion had a much faster degradation rate than the [PW12O40]3-anion in an excess of H2O2,which resulted in the formation of more catalytically active species. As a result,the lacunary-type phosphotungstate anion-based catalyst gave a better catalytic performance than the complete Keggin-type anion in ethyl acetate.
文摘For a prescribed set of lacunary data{(x_(ν),f_(ν),f^(″)_(ν)):ν=0,1,...,N}with equally spaced knot sequence in the unit interval,we show the existence of a fam-ily of fractal splines S^(α)_(b)∈C 3[0,1]satisfying S^(α)_(b)(x_(ν))=f_(ν),(S^(α)_(b))^(2)(x_(ν))=f^(″)_(ν)forν=0,1,...,N and suitable boundary conditions.To this end,the unique quintic spline introduced by A.Meir and A.Sharma[SIAM J.Numer.Anal.10(3)1973,pp.433-442]is generalized by using fractal functions with variable scaling pa-rameters.The presence of scaling parameters that add extra“degrees of freedom”,self-referentiality of the interpolant,and“fractality”of the third derivative of the in-terpolant are additional features in the fractal version,which may be advantageous in applications.If the lacunary data is generated from a functionΦsatisfying certain smoothness condition,then for suitable choices of scaling factors,the corresponding fractal spline S^(α)_(b)satisfies||Φ^(r)−(S^(α)_(b))(r)||∞→0 for 0≤r≤3,as the number of partition points increases.
基金Supported by the National Natural Science Foundation of China(Grant No.11301233)the Natural Science Foundation of Jiangxi Province(Grant No.20132BAB211002)the Youth Science Foundation of Education Bureau of Jiangxi Province(Grant No.GJJ14271)
文摘In this paper, we investigate the growth and value distribution of meromorphic solutions to higher order linear differential equations with some dominating coefficient being Lacunary series and the results of this paper improve and extend the previous results of J. Tu, 2013.
文摘A new heteropolycomplex, K 6H 3[ZnW 11 O 40 Al]·9 5H 2O was prepared and characterized by means of elemental analysis, IR, UV spectroscopy, 27 Al NMR , electrochemistry and X ray crystallography. The crystal of K 6H 3[ZnW 11 O 40 Al]·9 5H 2O is cubic, space group Fm 3m , with lattice constants a=b=c =2 144 8(2) nm, V =9 866(2) nm 3, Mo radiation, R =0 057 8 for 497 independent data with [ I>2δ(I) ]. The anion is of α type Keggin structure with C s symmetry.
基金sponsored by UGC, Goverment of India under the Major Research Project Scheme vide No. MRP-MAJOR-MATH-2013-29143
文摘In this paper, we have studied the Lacunary type of polynomials and proved a result which generalizes as well as refines some well-known polynomial inequalities regarding the growth of polynomials not vanishing inside a circle. Further the paper corrects the proofs of some already known results.
基金supported by the National Natural Science Foundation of China(21871025,21831001)the Recruitment Program of Global Experts(Young Talents)+1 种基金BIT Excellent Young Scholars Research Fundsupport from the Analysis and Testing Center of Beijing Instituted of Technology。
文摘Polyoxometalates(POMs)have been emerging as one of the most widely-studied interdisciplinary research fields of inorganic chemistry due to their structural diversities and versatile physicochemical properties.The incorporation of transition metals into lacunary POM ligands can lead to the formation of various transition-metal-substituted POMs(TMSPs)with wide potential applications in research areas including medicine,magnetism,functional materials,catalysis,and energy storage.In this review,we summarized the recent advances of mixed-transition-metal-substituted POMs(Mixed TMSPs)with respect to their synthetic strategies,structural diversities,and relevant applications.Finally,a comprehensive outlook of this research area was also prospected.
