Gradation of Algebras of Curves by the Winding Number

We construct a new grading on the Goldman Lie algebra of a closed oriented surface by the winding number. This grading induces a grading on the HOMFLY-PT skein algebra and related algebras. Our work supports the conjectures of B. Cooper and P. SamuelsonComment: Changed acknowledgments and Definition 2.

The sparse Blume-Emery-Griffiths model of associative memories

We analyze the Blume-Emery-Griffiths (BEG) associative memory with sparse patterns and at zero temperature. We give bounds on its storage capacity provided that we want the stored patterns to be fixed points of the retrieval dynamics. We compare our results to that of other models of sparse neural networks and show that the BEG model has a superior performance compared to them.Comment: 23 p

Series of Mixed Uranyl–Lanthanide (Ce, Nd) Organic Coordination Polymers with Aromatic Polycarboxylates Linkers

Three series of mixed uranyl-lanthanide (Ce or Nd) carboxylate coordination polymers have been successfully synthesized by means of a hydrothermal route using either conventional or microwave heating methods. These compounds have been prepared from mixtures of uranyl nitrate, lanthanide nitrate together with phthalic acid (<b>1</b>,<b>2</b>), pyromellitic acid (<b>3</b>,<b>4</b>), or mellitic acid (<b>5</b>,<b>6</b>) in aqueous solution. The X-ray diffraction (XRD) single-crystal revealed that the phthalate complex (UO₂)₄O₂Ln­(H₂O)₇(1,2-bdc)₄·NH₄·<i>x</i>H₂O (Ln = Ce­(<b>1</b>), Nd­(<b>2</b>); <i>x</i> = 1 for <b>1</b>, <i>x</i> = 0 for <b>2</b>), is based on the connection of tetranuclear uranyl-centered building blocks linked to discrete monomeric units LnO₂(H₂O)₇ via the organic species to generate infinite chains, intercalated by free ammonium cations. The pyromellitate phase (UO₂)₃Ln₂(H₂O)₁₂(btec)₃·5H₂O (Ce­(<b>3</b>), Nd­(<b>4</b>)) contains layers of monomeric uranyl-centered hexagonal and pentagonal bipyramids linked via the carboxylate arms of the organic molecules. The three-dimensionality of the structure is ensured by the connection of remaining free carboxylate groups with isolated monomeric units LnO₂(H₂O)₇. The network of the third series (UO₂)₂(OH)­Ln­(H₂O)₇(mel)·5H₂O (Ce­(<b>5</b>), Nd­(<b>6</b>)) is built up from dinuclear uranyl units forming layers through connection with the mellitate ligands, which are further linked to each other through discrete monomers LnO₃(H₂O)₆. The thermal decomposition of the various coordination complexes led to the formation of mixed uranium-lanthanide oxide, with the fluorite-type structure at 1500 °C (for <b>1</b>, <b>2</b>) or 1400 °C for <b>3</b>–<b>6</b>. Expected U/Ln ratio from the crystal structures were observed for compounds <b>1</b>–<b>6</b>

Uranyl and/or Rare-Earth Mellitates in Extended Organic–Inorganic Networks: A Unique Case of Heterometallic Cation–Cation Interaction with U^VIO–Ln^III Bonding (Ln = Ce, Nd)

A series of uranyl and lanthanide (trivalent Ce, Nd) mellitates (<i>mel</i>) has been hydrothermally synthesized in aqueous solvent. Mixtures of these 4f and 5f elements also revealed the formation of a rare case of lanthanide–uranyl coordination polymers. Their structures, determined by XRD single-crystal analysis, exhibit three distinct architectures. The pure lanthanide mellitate Ln₂(H₂O)₆(<i>mel</i>) possesses a 3D framework built up from the connection of isolated LnO₆(H₂O)₃ polyhedra (tricapped trigonal prism) through the mellitate ligand. The structure of the uranyl mellitate (UO₂)₃(H₂O)₆(<i>mel</i>)·11.5H₂O is lamellar and consists of 8-fold coordinated uranium atoms linked to each other through the organic ligand giving rise to the formation of a 2D 3⁶ net. The third structural type, (UO₂)₂Ln­(OH)­(H₂O)₃(<i>mel</i>)·2.5H₂O, involves direct oxygen bondings between the lanthanide and uranyl centers, with the isolation of a heterometallic dinuclear motif. The 9-fold coordinated Ln cation, LnO₅(OH)­(H₂O)₃, is linked to the 7-fold coordinated uranyl (UO₂)­O₄(OH) (pentagonal bipyramid) via one μ₂-hydroxo group and one μ₂-oxo group. The latter is shared between the uranyl bonding (UO = 1.777(4)/1.779(6) Å) and a long Ln–O bonding (Ce–O = 2.822(4) Å; Nd–O = 2.792(6) Å). This unusual linkage is a unique illustration of the so-called cation–cation interaction associating 4f and 5f metals. The dinuclear motif is then further connected through the mellitate ligand, and this generates organic–inorganic layers that are linked to each other via discrete uranyl (UO₂)­O₄ units (square bipyramid), which ensure the three-dimensional cohesion of the structure. The mixed U–Ln carboxylate is thermally decomposed from 260 to 280 °C and then transformed into the basic uranium oxide (U₃O₈) together with U–Ln oxide with the fluorite structural type (“(Ln,U)­O₂”). At 1400 °C, only fluorite type “(Ln,U)­O₂” is formed with the measured stoichiometry of U_0.63Ce_0.37O₂ and U_0.60Nd_0.40O_2−δ

Series of Mixed Uranyl–Lanthanide (Ce, Nd) Organic Coordination Polymers with Aromatic Polycarboxylates Linkers

Three series of mixed uranyl-lanthanide (Ce or Nd) carboxylate coordination polymers have been successfully synthesized by means of a hydrothermal route using either conventional or microwave heating methods. These compounds have been prepared from mixtures of uranyl nitrate, lanthanide nitrate together with phthalic acid (<b>1</b>,<b>2</b>), pyromellitic acid (<b>3</b>,<b>4</b>), or mellitic acid (<b>5</b>,<b>6</b>) in aqueous solution. The X-ray diffraction (XRD) single-crystal revealed that the phthalate complex (UO₂)₄O₂Ln­(H₂O)₇(1,2-bdc)₄·NH₄·<i>x</i>H₂O (Ln = Ce­(<b>1</b>), Nd­(<b>2</b>); <i>x</i> = 1 for <b>1</b>, <i>x</i> = 0 for <b>2</b>), is based on the connection of tetranuclear uranyl-centered building blocks linked to discrete monomeric units LnO₂(H₂O)₇ via the organic species to generate infinite chains, intercalated by free ammonium cations. The pyromellitate phase (UO₂)₃Ln₂(H₂O)₁₂(btec)₃·5H₂O (Ce­(<b>3</b>), Nd­(<b>4</b>)) contains layers of monomeric uranyl-centered hexagonal and pentagonal bipyramids linked via the carboxylate arms of the organic molecules. The three-dimensionality of the structure is ensured by the connection of remaining free carboxylate groups with isolated monomeric units LnO₂(H₂O)₇. The network of the third series (UO₂)₂(OH)­Ln­(H₂O)₇(mel)·5H₂O (Ce­(<b>5</b>), Nd­(<b>6</b>)) is built up from dinuclear uranyl units forming layers through connection with the mellitate ligands, which are further linked to each other through discrete monomers LnO₃(H₂O)₆. The thermal decomposition of the various coordination complexes led to the formation of mixed uranium-lanthanide oxide, with the fluorite-type structure at 1500 °C (for <b>1</b>, <b>2</b>) or 1400 °C for <b>3</b>–<b>6</b>. Expected U/Ln ratio from the crystal structures were observed for compounds <b>1</b>–<b>6</b>

Uranyl–Pyromellitate Coordination Polymers: Toward Three-Dimensional Open Frameworks with Large Channel Systems

Five coordination polymers based on uranyl–pyromellitates have been hydrothermally synthesized, and their single-crystal XRD structures have been analyzed. These different compounds, obtained with different ammonia concentrations, exhibit either three-dimensional (3D) or two-dimensional (2D) networks. Complex <b>1</b>, (UO₂)₂(H₂O)₂(btec)·H₂O, is a quite compact 3D structure containing isolated 7-fold coordinated uranyl cations linked through the pyromellitate (noted btec) and encapsulating free water species. Phase (NH₄)­[(UO₂)₂(OH)­(H₂O)­(btec)]·1.75H₂O (<b>2</b>) offers a second 3D architecture built up from dinuclear 7-fold coordinated uranyl units and mononuclear 8-fold coordinated uranyl units linked through the organic ligands. This framework is slightly more open because narrow one-dimensional (1D) channels trapping water species are visible. Phase <b>3</b>, (NH₄)₂[(UO₂)₆O₂(OH)₄(btec)_1.5]·11H₂O, consists of large 1D lozenge-shaped channels (8.2 Å × 8.6 Å) delimited by infinite ribbons (composed of 7-fold coordinated uranyl polyhedra sharing edges) and pyromellitate ligands. Ammonium cations as well as water molecules are trapped within the channels. The fourth compound, (NH₄)₆[(UO₂)₃(btec)₃]·12H₂O (<b>4</b>), is lamellar with sheets containing 8-fold coordinated uranyl centers linked through the btec molecules, which have two nonbonded carboxylate functions interacting with the intercalated ammonium cations. Compound <b>5</b> also consists of a layered structure, (UO₂)₃(OH)₂(H₂O)₂(btec), with uncommon trinuclear building blocks containing 7-fold (×2) and 8-fold (×1) coordinated uranyl centers, linked through the btec molecules. Fluorescence spectra of compounds <b>1</b>, <b>3</b>, and <b>4</b> are also measured

Uranyl–Pyromellitate Coordination Polymers: Toward Three-Dimensional Open Frameworks with Large Channel Systems

Five coordination polymers based on uranyl–pyromellitates have been hydrothermally synthesized, and their single-crystal XRD structures have been analyzed. These different compounds, obtained with different ammonia concentrations, exhibit either three-dimensional (3D) or two-dimensional (2D) networks. Complex <b>1</b>, (UO₂)₂(H₂O)₂(btec)·H₂O, is a quite compact 3D structure containing isolated 7-fold coordinated uranyl cations linked through the pyromellitate (noted btec) and encapsulating free water species. Phase (NH₄)­[(UO₂)₂(OH)­(H₂O)­(btec)]·1.75H₂O (<b>2</b>) offers a second 3D architecture built up from dinuclear 7-fold coordinated uranyl units and mononuclear 8-fold coordinated uranyl units linked through the organic ligands. This framework is slightly more open because narrow one-dimensional (1D) channels trapping water species are visible. Phase <b>3</b>, (NH₄)₂[(UO₂)₆O₂(OH)₄(btec)_1.5]·11H₂O, consists of large 1D lozenge-shaped channels (8.2 Å × 8.6 Å) delimited by infinite ribbons (composed of 7-fold coordinated uranyl polyhedra sharing edges) and pyromellitate ligands. Ammonium cations as well as water molecules are trapped within the channels. The fourth compound, (NH₄)₆[(UO₂)₃(btec)₃]·12H₂O (<b>4</b>), is lamellar with sheets containing 8-fold coordinated uranyl centers linked through the btec molecules, which have two nonbonded carboxylate functions interacting with the intercalated ammonium cations. Compound <b>5</b> also consists of a layered structure, (UO₂)₃(OH)₂(H₂O)₂(btec), with uncommon trinuclear building blocks containing 7-fold (×2) and 8-fold (×1) coordinated uranyl centers, linked through the btec molecules. Fluorescence spectra of compounds <b>1</b>, <b>3</b>, and <b>4</b> are also measured

Uranyl–Pyromellitate Coordination Polymers: Toward Three-Dimensional Open Frameworks with Large Channel Systems

Five coordination polymers based on uranyl–pyromellitates have been hydrothermally synthesized, and their single-crystal XRD structures have been analyzed. These different compounds, obtained with different ammonia concentrations, exhibit either three-dimensional (3D) or two-dimensional (2D) networks. Complex <b>1</b>, (UO₂)₂(H₂O)₂(btec)·H₂O, is a quite compact 3D structure containing isolated 7-fold coordinated uranyl cations linked through the pyromellitate (noted btec) and encapsulating free water species. Phase (NH₄)­[(UO₂)₂(OH)­(H₂O)­(btec)]·1.75H₂O (<b>2</b>) offers a second 3D architecture built up from dinuclear 7-fold coordinated uranyl units and mononuclear 8-fold coordinated uranyl units linked through the organic ligands. This framework is slightly more open because narrow one-dimensional (1D) channels trapping water species are visible. Phase <b>3</b>, (NH₄)₂[(UO₂)₆O₂(OH)₄(btec)_1.5]·11H₂O, consists of large 1D lozenge-shaped channels (8.2 Å × 8.6 Å) delimited by infinite ribbons (composed of 7-fold coordinated uranyl polyhedra sharing edges) and pyromellitate ligands. Ammonium cations as well as water molecules are trapped within the channels. The fourth compound, (NH₄)₆[(UO₂)₃(btec)₃]·12H₂O (<b>4</b>), is lamellar with sheets containing 8-fold coordinated uranyl centers linked through the btec molecules, which have two nonbonded carboxylate functions interacting with the intercalated ammonium cations. Compound <b>5</b> also consists of a layered structure, (UO₂)₃(OH)₂(H₂O)₂(btec), with uncommon trinuclear building blocks containing 7-fold (×2) and 8-fold (×1) coordinated uranyl centers, linked through the btec molecules. Fluorescence spectra of compounds <b>1</b>, <b>3</b>, and <b>4</b> are also measured

Uranyl–Pyromellitate Coordination Polymers: Toward Three-Dimensional Open Frameworks with Large Channel Systems

Five coordination polymers based on uranyl–pyromellitates have been hydrothermally synthesized, and their single-crystal XRD structures have been analyzed. These different compounds, obtained with different ammonia concentrations, exhibit either three-dimensional (3D) or two-dimensional (2D) networks. Complex <b>1</b>, (UO₂)₂(H₂O)₂(btec)·H₂O, is a quite compact 3D structure containing isolated 7-fold coordinated uranyl cations linked through the pyromellitate (noted btec) and encapsulating free water species. Phase (NH₄)­[(UO₂)₂(OH)­(H₂O)­(btec)]·1.75H₂O (<b>2</b>) offers a second 3D architecture built up from dinuclear 7-fold coordinated uranyl units and mononuclear 8-fold coordinated uranyl units linked through the organic ligands. This framework is slightly more open because narrow one-dimensional (1D) channels trapping water species are visible. Phase <b>3</b>, (NH₄)₂[(UO₂)₆O₂(OH)₄(btec)_1.5]·11H₂O, consists of large 1D lozenge-shaped channels (8.2 Å × 8.6 Å) delimited by infinite ribbons (composed of 7-fold coordinated uranyl polyhedra sharing edges) and pyromellitate ligands. Ammonium cations as well as water molecules are trapped within the channels. The fourth compound, (NH₄)₆[(UO₂)₃(btec)₃]·12H₂O (<b>4</b>), is lamellar with sheets containing 8-fold coordinated uranyl centers linked through the btec molecules, which have two nonbonded carboxylate functions interacting with the intercalated ammonium cations. Compound <b>5</b> also consists of a layered structure, (UO₂)₃(OH)₂(H₂O)₂(btec), with uncommon trinuclear building blocks containing 7-fold (×2) and 8-fold (×1) coordinated uranyl centers, linked through the btec molecules. Fluorescence spectra of compounds <b>1</b>, <b>3</b>, and <b>4</b> are also measured

Series of Mixed Uranyl–Lanthanide (Ce, Nd) Organic Coordination Polymers with Aromatic Polycarboxylates Linkers

Three series of mixed uranyl-lanthanide (Ce or Nd) carboxylate coordination polymers have been successfully synthesized by means of a hydrothermal route using either conventional or microwave heating methods. These compounds have been prepared from mixtures of uranyl nitrate, lanthanide nitrate together with phthalic acid (<b>1</b>,<b>2</b>), pyromellitic acid (<b>3</b>,<b>4</b>), or mellitic acid (<b>5</b>,<b>6</b>) in aqueous solution. The X-ray diffraction (XRD) single-crystal revealed that the phthalate complex (UO₂)₄O₂Ln­(H₂O)₇(1,2-bdc)₄·NH₄·<i>x</i>H₂O (Ln = Ce­(<b>1</b>), Nd­(<b>2</b>); <i>x</i> = 1 for <b>1</b>, <i>x</i> = 0 for <b>2</b>), is based on the connection of tetranuclear uranyl-centered building blocks linked to discrete monomeric units LnO₂(H₂O)₇ via the organic species to generate infinite chains, intercalated by free ammonium cations. The pyromellitate phase (UO₂)₃Ln₂(H₂O)₁₂(btec)₃·5H₂O (Ce­(<b>3</b>), Nd­(<b>4</b>)) contains layers of monomeric uranyl-centered hexagonal and pentagonal bipyramids linked via the carboxylate arms of the organic molecules. The three-dimensionality of the structure is ensured by the connection of remaining free carboxylate groups with isolated monomeric units LnO₂(H₂O)₇. The network of the third series (UO₂)₂(OH)­Ln­(H₂O)₇(mel)·5H₂O (Ce­(<b>5</b>), Nd­(<b>6</b>)) is built up from dinuclear uranyl units forming layers through connection with the mellitate ligands, which are further linked to each other through discrete monomers LnO₃(H₂O)₆. The thermal decomposition of the various coordination complexes led to the formation of mixed uranium-lanthanide oxide, with the fluorite-type structure at 1500 °C (for <b>1</b>, <b>2</b>) or 1400 °C for <b>3</b>–<b>6</b>. Expected U/Ln ratio from the crystal structures were observed for compounds <b>1</b>–<b>6</b>