Robust Stackelberg Equilibria in Extensive-Form Games and Extension to Limited Lookahead

Stackelberg equilibria have become increasingly important as a solution concept in computational game theory, largely inspired by practical problems such as security settings. In practice, however, there is typically uncertainty regarding the model about the opponent. This paper is, to our knowledge, the first to investigate Stackelberg equilibria under uncertainty in extensive-form games, one of the broadest classes of game. We introduce robust Stackelberg equilibria, where the uncertainty is about the opponent's payoffs, as well as ones where the opponent has limited lookahead and the uncertainty is about the opponent's node evaluation function. We develop a new mixed-integer program for the deterministic limited-lookahead setting. We then extend the program to the robust setting for Stackelberg equilibrium under unlimited and under limited lookahead by the opponent. We show that for the specific case of interval uncertainty about the opponent's payoffs (or about the opponent's node evaluations in the case of limited lookahead), robust Stackelberg equilibria can be computed with a mixed-integer program that is of the same asymptotic size as that for the deterministic setting.Comment: Published at AAAI1

Mimic Carbonic Anhydrase Using Metal–Organic Frameworks for CO₂ Capture and Conversion

Carbonic anhydrase (CA) is a zinc-containing metalloprotein, in which the Zn active center plays the key role to transform CO₂ into carbonate. Inspired by nature, herein we used metal–organic frameworks (MOFs) to mimic CA for CO₂ conversion, on the basis of the structural similarity between the Zn coordination in MOFs and CA active center. The biomimetic activity of MOFs was investigated by detecting the hydrolysis of <i>para</i>-nitrophenyl acetate, which is a model reaction used to evaluate CA activity. The biomimetic materials (e.g., CFA-1) showed good catalytic activity, and excellent reusability, and solvent and thermal stability, which is very important for practical applications. In addition, ZIF-100 and CFA-1 were used to mimic CA to convert CO₂ gas, and exhibited good efficiency on CO₂ conversion compared with those of other porous materials (e.g., MCM-41, active carbon). This biomimetic study revealed a novel CO₂ treatment method. Instead of simply using MOFs to absorb CO₂, ZIF-100 and CFA-1 were used to mimic CA for in situ CO₂ conversion, which provides a new prospect in the biological and industrial applications of MOFs

Three Porphyrin-Encapsulating Metal–Organic Materials with Ordered Metalloporphyrin Moieties

1,3,5-benzenetricarboxylate, biphenyl-3,4′,5-tricarboxylate, and [1,1′:3′,1″-terphenyl]-4,4″,5′-tricarboxylate react with Cd­(II) cations in the presence of meso-tetra­(<i>N</i>-methyl-4-pyridyl) porphine cations to afford three metalloporphyrin-encapsulating metal–organic materials, <b>porph@MOM-11</b>, <b>-12</b> and <b>-13</b>, respectively. The metalloporphyrin moieties are ordered within channels and closely fit hexagonal or rectangular cavities, thereby facilitating a better understanding of the structure-directing effect that can be promoted by metalloporphyrins. <b>Porph@MOM-12</b> is noteworthy because it exhibits two distinct types of hexagonal channels and it represents the first example of a net that exhibits <b>mzz</b> topology

Toward “metalloMOFzymes”: Metal–Organic Frameworks with Single-Site Metal Catalysts for Small-Molecule Transformations

Metal–organic frameworks (MOFs) are being increasingly studied as scaffolds and supports for catalysis. The solid-state structures of MOFs, combined with their high porosity, suggest that MOFs may possess advantages shared by both heterogeneous and homogeneous catalysts, with few of the shortcomings of either. Herein, efforts to create single-site catalytic metal centers appended to the organic ligand struts of MOFs will be discussed. Reactions important for advanced energy applications, such as H₂ production and CO₂ reduction, will be highlighted. Examining how these active sites can be introduced, their performance, and their existing limitations should provide direction for design of the next generation of MOF-based catalysts for energy-relevant, small-molecule transformations. Finally, the introduction of second-sphere interactions (e.g., hydrogen bonding via squaramide groups) as a possible route to enhancing the activity of these metal centers is reported

Stepwise Transformation of the Molecular Building Blocks in a Porphyrin-Encapsulating Metal–Organic Material

When immersed in solutions containing Cu­(II) cations, the microporous metal–organic material <b>P11</b> ([Cd₄(BPT)₄]·[Cd­(C₄₄H₃₆N₈)­(S)]·[S], BPT = biphenyl-3,4′,5-tricarboxylate) undergoes a transformation of its [Cd₂(COO)₆]^2– molecular building blocks (MBBs) into novel tetranuclear [Cu₄X₂(COO)₆(S)₂] MBBs to form <b>P11-Cu</b>. The transformation occurs in single-crystal to single-crystal fashion, and its stepwise mechanism was studied by varying the Cd²⁺/Cu²⁺ ratio of the solution in which crystals of <b>P11</b> were immersed. <b>P11-16/1</b> (Cd in framework retained, Cd in encapsulated porphyrins exchanged) and other intermediate phases were thereby isolated and structurally characterized. <b>P11-16/1</b> and <b>P11-Cu</b> retain the microporosity of <b>P11</b>, and the relatively larger MBBs in <b>P11-Cu</b> permit a 20% unit cell expansion and afford a higher surface area and a larger pore size

Stepwise Transformation of the Molecular Building Blocks in a Porphyrin-Encapsulating Metal–Organic Material

When immersed in solutions containing Cu­(II) cations, the microporous metal–organic material <b>P11</b> ([Cd₄(BPT)₄]·[Cd­(C₄₄H₃₆N₈)­(S)]·[S], BPT = biphenyl-3,4′,5-tricarboxylate) undergoes a transformation of its [Cd₂(COO)₆]^2– molecular building blocks (MBBs) into novel tetranuclear [Cu₄X₂(COO)₆(S)₂] MBBs to form <b>P11-Cu</b>. The transformation occurs in single-crystal to single-crystal fashion, and its stepwise mechanism was studied by varying the Cd²⁺/Cu²⁺ ratio of the solution in which crystals of <b>P11</b> were immersed. <b>P11-16/1</b> (Cd in framework retained, Cd in encapsulated porphyrins exchanged) and other intermediate phases were thereby isolated and structurally characterized. <b>P11-16/1</b> and <b>P11-Cu</b> retain the microporosity of <b>P11</b>, and the relatively larger MBBs in <b>P11-Cu</b> permit a 20% unit cell expansion and afford a higher surface area and a larger pore size

Stepwise Transformation of the Molecular Building Blocks in a Porphyrin-Encapsulating Metal–Organic Material

When immersed in solutions containing Cu­(II) cations, the microporous metal–organic material <b>P11</b> ([Cd₄(BPT)₄]·[Cd­(C₄₄H₃₆N₈)­(S)]·[S], BPT = biphenyl-3,4′,5-tricarboxylate) undergoes a transformation of its [Cd₂(COO)₆]^2– molecular building blocks (MBBs) into novel tetranuclear [Cu₄X₂(COO)₆(S)₂] MBBs to form <b>P11-Cu</b>. The transformation occurs in single-crystal to single-crystal fashion, and its stepwise mechanism was studied by varying the Cd²⁺/Cu²⁺ ratio of the solution in which crystals of <b>P11</b> were immersed. <b>P11-16/1</b> (Cd in framework retained, Cd in encapsulated porphyrins exchanged) and other intermediate phases were thereby isolated and structurally characterized. <b>P11-16/1</b> and <b>P11-Cu</b> retain the microporosity of <b>P11</b>, and the relatively larger MBBs in <b>P11-Cu</b> permit a 20% unit cell expansion and afford a higher surface area and a larger pore size

Template-Directed Synthesis of Nets Based upon Octahemioctahedral Cages That Encapsulate Catalytically Active Metalloporphyrins

<i>meso</i>-Tetra­(<i>N</i>-methyl-4-pyridyl)­porphine tetratosylate (TMPyP) templates the synthesis of six new metal–organic materials by the reaction of benzene-1,3,5-tricarboxylate with transition metals, five of which exhibit HKUST-1 or <b>tbo</b> topology (M = Fe, Mn, Co, Ni, Mg). The resulting materials, <b>porph@MOMs</b>, selectively encapsulate the corresponding metalloporphyrins in octahemioctahedral cages and can serve as size-selective heterogeneous catalysts for oxidation of olefins

Template-Directed Synthesis of Nets Based upon Octahemioctahedral Cages That Encapsulate Catalytically Active Metalloporphyrins

<i>meso</i>-Tetra­(<i>N</i>-methyl-4-pyridyl)­porphine tetratosylate (TMPyP) templates the synthesis of six new metal–organic materials by the reaction of benzene-1,3,5-tricarboxylate with transition metals, five of which exhibit HKUST-1 or <b>tbo</b> topology (M = Fe, Mn, Co, Ni, Mg). The resulting materials, <b>porph@MOMs</b>, selectively encapsulate the corresponding metalloporphyrins in octahemioctahedral cages and can serve as size-selective heterogeneous catalysts for oxidation of olefins

Template-Directed Synthesis of Nets Based upon Octahemioctahedral Cages That Encapsulate Catalytically Active Metalloporphyrins

<i>meso</i>-Tetra­(<i>N</i>-methyl-4-pyridyl)­porphine tetratosylate (TMPyP) templates the synthesis of six new metal–organic materials by the reaction of benzene-1,3,5-tricarboxylate with transition metals, five of which exhibit HKUST-1 or <b>tbo</b> topology (M = Fe, Mn, Co, Ni, Mg). The resulting materials, <b>porph@MOMs</b>, selectively encapsulate the corresponding metalloporphyrins in octahemioctahedral cages and can serve as size-selective heterogeneous catalysts for oxidation of olefins