Über die optimale Beleuchtung einer geraden Straße

The following problem is due toL. Fejes Tóth: For a given sort of lamps letf(x) be the intensity of brightness in each point having distancex from the foot of the lamp. How must infinitely many lamps of this kind be arranged on a both-side infinite rectilinear road, such that the infimum of the intensities of brightness, extended over the whole street, is maximal, subject to the condition that the average number of lamps per kilometer is bounded by a fixed number (“Best distributions”)? The main result of this paper is: Iff is strictly convex, the equidistant distribution is best.     

On the Fourier transformation on spaces of (p,q)-multipliers

LetG denote a locally compact abelian topological group. The aim of the present paper is to prove an “intermediate” result between two well-known results ofL. Hörmander andG. I. Gaudry concerning the structure of the spaces ?_G?μ?t ^p,q (G).     

Eine spezielle Vorgangsdauerverteilung zur Netzplananalyse durch Simulation

Zusammenfassung Es wird eineC genannte Verteilung bereitgestellt, die sich zur Erfassung von Vorgangsdauern bei Netzplanproblemen eignet und durch besondere Wirtschaftlichkeit bei Simulationsmethoden auszeichnet.

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Summary In this paper a distributionC will be presented for the activity duration times in network problems which from the economical point of view is fit for the solution to network problems via Monte Carlo simulation.

     

Zur Entscheidungstheorie bei mehrfacher Zielsetzung

Zusammenfassung Der Begriff eines rationalen Lösunxgsprinzips für Entscheidungsprobleme mit mehrfacher Zielsetzung wird definitorisch eingeführt. Anhand der definierenden Bedingungen werden die neueren Lösungsansätze auf Rationalität untersucht. Zusätzlich wird besonderes Gewicht auf die kritische Untersuchung der den einzelnen Verfahren zugrundeliegenden Informationsannahmen gelegt. Dabei zeigt sich, wie auch aus empirischen Untersuchungen abgeleitet wird, daß es bei der Formulierung von Lösungsansätzen auf eine präzise Darstellung des Lösungsprinzips und der damit im Zusammenhang stehenden Informationsannahme ankommt; erst auf dieser Grundlage kann die entscheidungspraktische Relevanz solcher Verfahren überprüft werden.

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Summary For multiobjective decision problems a notion of a rational solution principle is introduced by definition. The recently suggested solution approaches are tested for rationality under the defining conditions. Additionally, great importance is attached to the critical analysis of the information requirements underlying the different methods. It turns out, as well as it will be derived from empirical investigations, that a precise presentation of the solution principle and of the corresponding information requirements is indispensable formulating such approaches; first on this basis the practical relevance of solution procedures can be examined.

     

Ligand-promoted palladium-catalyzed β-methylene C–H arylation of primary aldehydes

The transient directing group (TDG) strategy allowed long awaited access to the direct β-C(sp³)–H functionalization of unmasked aliphatic aldehydes via palladium catalysis. However, the current techniques are restricted to terminal methyl functionalization, limiting their structural scopes and applicability. Herein, we report the development of a direct Pd-catalyzed methylene β-C–H arylation of linear unmasked aldehydes by using 3-amino-3-methylbutanoic acid as a TDG and 2-pyridone as an external ligand. Density functional theory calculations provided insights into the reaction mechanism and shed light on the roles of the external and transient directing ligands in the catalytic transformation.

Aliphatic aldehydes are among the most common structural units in organic and medicinal chemistry research. Direct C–H functionalization has enabled efficient and site-selective derivatization of aliphatic aldehydes.

Simple aliphatic functional groups enrich the skeletal backbones of many natural products, pharmaceuticals, and other industrial materials, influencing the utility and applications of these substances and dictating their reactivity and synthetic modification pathways. Aliphatic aldehydes are some of the most ubiquitous structural units in organic materials.¹ Their relevance in nature and industry alike, combined with their reactivity and synthetic versatility, attracted much attention from the synthetic organic and medicinal chemistry communities over the years (Fig. 1).² Efficient means to the functionalization of these molecules have always been highly sought after.Open in a separate windowFig. 1Select aliphatic aldehyde-containing medicines and biologically active molecules.Traditionally, scientists have utilized the high reactivity of the aldehyde moiety in derivatizing a variety of functional groups by the means of red-ox and nucleophilic addition reactions. The resourceful moiety was also notoriously used to install functional groups at the α-position via condensation and substitution pathways.³ Although β-functionalization is just as robust, it has generally been more restrictive as it often requires the use of α,β-unsaturated aldehydes.^4,5 Hence, transition metal catalysis emerged as a powerful tool to access β-functionalization in saturated aldehydes.⁶ Most original examples of metal-catalyzed β-C–H functionalization of aliphatic aldehydes required the masking of aldehydes into better metal coordinating units since free unmasked aldehydes could not form stable intermediates with metals like palladium on their own.⁷ Although the masking of the aldehyde moiety into an oxime, for example, enabled the formation of stable 5-membered palladacycles, affording β-functionalized products, this system requires the installation of the directing group prior to the functionalization, as well as the subsequent unmasking upon the reaction completion, compromising the step economy and atom efficiency of the overall process.⁸ Besides, some masking and unmasking protocols might not be compatible with select substrates, especially ones rich in functional groups. As a result, the development of a one-step direct approach to the β-C–H functionalization of free aliphatic aldehydes was a demanding target for synthetic chemists.α-Amino acids have been demonstrated as effective transient directing groups (TDGs) in the remote functionalization of o-alkyl benzaldehydes and aliphatic ketones by the Yu group in 2016.⁹ Shortly after, our group disclosed the first report on the direct β-C–H arylation of aliphatic aldehydes using 3-aminopropanoic acid or 3-amino-3-methylbutanoic acid as a TDG.¹⁰ The TDG was found to play a similar role to that of the oxime directing group by binding to the substrate via reversible imine formation, upon which, it assists in the assembly of a stable palladacycle, effectively functionalizing the β-position.¹¹ Since the binding of the TDG is reversible and temporary, it is automatically removed upon functionalization, yielding an efficient and step-economic transformation. This work was succeeded by many other reports that expanded the reaction and the TDG scopes.^12–14 However, this system suffers from a significant restriction that demanded resolution; only substitution of methyl C–H bonds of linear aldehydes was made possible via this approach (Scheme 1a–e). The steric limitations caused by incorporating additional groups at the β-carbon proved to compromise the formation of the palladacycle intermediate, rendering the subsequent functionalization a difficult task.¹²Open in a separate windowScheme 1Pd-catalyzed β-C–H bond functionalization of aliphatic aldehydes enabled by transient directing groups.Encouraged by the recent surge in use of 2-pyridone ligands to stabilize palladacycle intermediates,^15,16 we have successfully developed the first example of TDG-enabled Pd-catalyzed methylene β-C–H arylation in primary aldehydes via the assistance of 2-pyridones as external ligands (Scheme 1f). The incorporation of 2-pyridones proved to lower the activation energy of the C–H bond cleavage, promoting the formation of the intermediate palladacycles even in the presence of relatively bulky β-substituents.¹⁷ This key advancement significantly broadens the structural scopes and applications of this process and promises future asymmetric possibilities, perhaps via the use of a chiral TDG or external ligand or both. Notably, a closely related work from Yu''s group was published at almost the same time.¹⁸We commenced our investigation of the reaction parameters by employing n-pentanal (1a) as an unbiased linear aldehyde and 4-iodoanisole (2a) in the presence of catalytic Pd(OAc)₂ and stoichiometric AgTFA, alongside 3-amino-3-methylbutanoic acid (TDG1) and 3-(trifluoromethyl)-5-nitropyridin-2-ol (L1) at 100 °C (ii) sources proved Pd(OAc)₂ to be the optimal catalyst, while Pd(TFA)₂, PdCl₂ and PdBr₂ provided only moderate yields (entries 10–12). Notably, a significantly lower yield was observed in the absence of the 2-pyridone ligand, and no desired product was isolated altogether in the absence of the TDG (entries 13 and 14). The incorporation of 15 mol% Pd catalyst was deemed necessary after only 55% yield of 3a was obtained when 10 mol% loading of Pd(OAc)₂ was instead used (entry 15).Optimization of reaction conditions^a

Entry	Pd source	L (mol%)	TDG1 (mol%)	Solvent (v/v, mL)	Yield (%)
1	Pd(OAc)2	L1 (30)	TDG1 (40)	HFIP	30
2	Pd(OAc)2	L1 (30)	TDG1 (40)	AcOH	<5
3	Pd(OAc)2	L1 (30)	TDG1 (40)	HFIP/AcOH (1 : 1)	28
4	Pd(OAc)2	L1 (30)	TDG1 (40)	HFIP/AcOH (9 : 1)	47
5	Pd(OAc)2	L1 (30)	TDG1 (40)	HFIP/AcOH (1 : 9)	<5
6	Pd(OAc)2	L1 (30)	TDG1 (60)	HFIP/AcOH (9 : 1)	50
7	Pd(OAc)2	L1 (30)	TDG1 (80)	HFIP/AcOH (9 : 1)	25
8	Pd(OAc)2	L1 (60)	TDG1 (60)	HFIP/AcOH (9 : 1)	70(68)b
9	Pd(OAc)2	L1 (75)	TDG1 (60)	HFIP/AcOH (9 : 1)	51
10	Pd(TFA)2	L1 (60)	TDG1 (60)	HFIP/AcOH (9 : 1)	60
11	PdCl2	L1 (60)	TDG1 (60)	HFIP/AcOH (9 : 1)	52
12	PdBr2	L1 (60)	TDG1 (60)	HFIP/AcOH (9 : 1)	54
13	Pd(OAc)2	—	TDG1 (60)	HFIP/AcOH (9 : 1)	9
14	Pd(OAc)2	L1 (60)	—	HFIP/AcOH (9 : 1)	0
15c	Pd(OAc)2	L1 (60)	TDG1 (60)	HFIP/AcOH (9 : 1)	55

Open in a separate window^aReaction conditions: 1a (0.2 mmol), 2a (0.4 mmol), Pd source (15 mol%), AgTFA (0.3 mmol), L1, TDG1, solvent (2.0 mL), 100 °C, 12 h. Yields are based on 1a, determined by ¹H-NMR using dibromomethane as an internal standard.^bIsolated yield.^cPd(OAc)₂ (10 mol%).To advance our optimization of the reaction conditions, a variety of 2-pyridones and TDGs were tested (Scheme 2). Originally, pyridine-2(1H)-one (L2) was examined as the external ligand, but it only yielded the product (3a) in 7% NMR yield. Similarly, other mono- and di-substituted 2-pyridone ligands (L3–L10) also produced low yields, fixating L1 as the optimal external ligand. Next, various α- and β-amino acids (TDG1–10) were evaluated, yet TDG1 persisted as the optimal transient directing group. These amino acid screening results also suggest that a [5,6]-bicyclic palladium species is likely the key intermediate in this protocol since only β-amino acids were found to provide appreciable yields, whereas α-amino acids failed to yield more than trace amounts of the product. The supremacy of TDG1 when compared to other β-amino acids is presumably due to the Thorpe–Ingold effect that perhaps helps facilitate the C–H bond cleavage and stabilize the [5,6]-bicyclic intermediate further.Open in a separate windowScheme 2Optimization of 2-pyridone ligands and transient directing groups.With the optimized reaction conditions in hand, substrate scope study of primary aliphatic aldehydes was subsequently carried out (Scheme 3). A variety of linear primary aliphatic aldehydes bearing different chain lengths provided the corresponding products 3a–e in good yields. Notably, relatively sterically hindered methylene C–H bonds were also functionalized effectively (3f and 3g). Additionally, 4-phenylbutanal gave rise to the desired product 3h in a highly site-selective manner, suggesting that functionalization of the methylene β-C–H bond is predominantly favored over the more labile benzylic C–H bond. It is noteworthy that the amide group was also well-tolerated and the desired product 3j was isolated in 60% yield. As expected, with n-propanal as the substrate, β-mono- (3k1) and β,β-disubstituted products (3k2) were isolated in 22% and 21% yields respectively. However, in the absence of the key external 2-pyridone ligand, β-monosubstituted product (3k1) was obtained exclusively, albeit with a low yield, indicating preference for functionalizing the β-C(sp³)–H bond of the methyl group over the benzylic methylene group.Open in a separate windowScheme 3Scope of primary aliphatic aldehydes. Reaction conditions: 1 (0.2 mmol), 2a (0.4 mmol), Pd(OAc)₂ (15 mol%), AgTFA (0.3 mmol), L1 (60 mol%), TDG1 (60 mol%), HFIP (1.8 mL), HOAc (0.2 mL), 100 °C, 12 h. Isolated yields. ^aL1 (60 mol%) was absent and yields are given in parentheses.Next, substrate scope study on aryl iodides was surveyed (Scheme 4). Iodobenzenes bearing either an electron-donating or electron-withdrawing group at the para-, meta-, or ortho-position were all found compatible with our catalytic system (3l–3ah). Surprisingly, ortho-methyl- and fluoro-substituted aryl iodides afforded the products in only trace amounts. However, aryl iodide with ortho-methoxy group provided the desired product 3ac in a moderate yield. Notably, a distinctive electronic effect pattern was not observed in the process. It should be mentioned that arylated products bearing halogen, ester, and cyano groups could be readily converted to other molecules, which significantly improves the synthetic applicability of the process. Delightfully, aryl iodide-containing natural products like ketoprofen, fenchol and menthol were proven compatible, supplying the corresponding products in moderate yields. Unfortunately, (hetero)aryl iodides including 2-iodopyridine, 3-iodopyridine, 4-iodopyridine and 4-iodo-2-chloropyridine failed to produce the corresponding products. Although our protocol provides a novel and direct pathway to construct β-arylated primary aliphatic aldehydes, the yields of most examples are modest. The leading reasons for this compromise are the following: (1) aliphatic aldehydes are easily decomposed or oxidized to acids; (2) some of the prepared β-arylated aldehyde products may be further transformed into the corresponding α,β-unsaturated aldehydes.Open in a separate windowScheme 4Scope of aryl iodides. Reaction conditions: 1a (0.2 mmol), 2 (0.4 mmol), Pd(OAc)₂ (15 mol%), AgTFA (0.3 mmol), L1 (60 mol%), TDG1 (60 mol%), HFIP (1.8 mL), HOAc (0.2 mL), 100 °C, 12 h. Isolated yields.Density functional theory (DFT) calculations were performed to help investigate the reaction mechanism and to elucidate the role of the ligand in improving the reactivity (Fig. 2). The condensation of the aliphatic aldehyde 1a with the TDG to form imine-1a was found thermodynamically neutral (ΔG° = −0.1 kcal mol⁻¹). As a result, it was permissible to use imine-1a directly in the calculations. According to the calculations results, the precatalyst [Pd(OAc)₂]₃, a trimeric complex, initially experiences dissociation and ligand metathesis with imine-1a to generate the Pd(ii) intermediate IM1, which is thermodynamically favorable by 21.9 kcal mol⁻¹. Consequently, the deprotonated imine-1a couples to the bidentate ligand to form the stable six-membered chelate complex IM1. Therefore, IM1 is indeed the catalyst resting state and serves as the zero point to the energy profile. We have identified two competitive pathways for the Pd(ii)-catalyzed C–H activation starting from IM1, one of which incorporates L1 and another which does not. On the one hand, an acetate ligand substitutes one imine-1a chelator in IM1 to facilitate the subsequent C–H activation leading to IM2, which undergoes C(sp³)–H activation through concerted metalation-deprotonation (CMD) viaTS1 (ΔG^‡ = 37.4 kcal mol⁻¹). However, this kinetic barrier is thought to be too high to account for the catalytic activity at 100 °C. On the other hand, the chelate imine-1a could be replaced by two N-coordinated ligands (L1) leading to the Pd(ii) complex IM3. This process is endergonic by 6.4 kcal mol⁻¹. To allow the ensuing C–H activation, IM3 dissociates one ligand (L1) producing the active species IM4, which undergoes TS2 to cleave the β-C(sp³)–H bond and form the [5,6]-bicyclic Pd(ii) intermediate IM5. Although this step features an energy barrier of 31.2 kcal mol⁻¹, it is thought to be feasible under the experimental conditions (100 °C). Possessing similar coordination ability to that of pyridine, the ligand (L1) effectively stabilizes the Pd(ii) center in the C–H activation process, indicating that this step most likely involves a manageable kinetic barrier. This result explicates the origin of the ligand-enabled reactivity (TS2vs.TS1). Additionally, we considered the γ-C(sp³)–H activation pathway viaTS2′ which was found to have a barrier of 35.5 kcal mol⁻¹. The higher energy barrier of TS2′ compared to that of TS2 is attributed to its larger ring strain in the [6,6]-bicyclic Pd(ii) transition state, which reveals the motive for the site-selectivity. Reverting back to the supposed pathway, upon the formation of the bicyclic intermediate IM5, it undergoes ligand/substrate replacement to afford intermediate IM6, at which the Ar–I coordinates to the Pd(ii) center to enable oxidative addition viaTS3 (ΔG^‡ = 27.4 kcal mol⁻¹) leading to the five-coordinate Pd(iv) complex IM7. Undergoing direct C–C reductive elimination in IM7 would entail a barrier of 29.6 kcal mol⁻¹ (TS4). Alternatively, iodine abstraction by the silver(i) salt in IM7 is thermodynamically favorable and irreversible, yielding the Pd(iv) intermediate IM8 coordinated to a TFA ligand. Subsequently, C–C reductive coupling viaTS5 generates the Pd(ii) complex IM9 and concludes the arylation process. This step was found both kinetically facile (6.1 kcal mol⁻¹) and thermodynamically favorable (30.7 kcal mol⁻¹). Finally, IM9 reacts with imine-1avia metathesis to regenerate the palladium catalyst IM1 and release imine-3a in a highly exergonic step (21.0 kcal mol⁻¹). Ultimately, imine-3a undergoes hydrolysis to yield the aldehyde product 3a and to release the TDG.Open in a separate windowFig. 2Free energy profiles for the ligand-promoted Pd(ii)-catalyzed site-selective C–H activation and C–C bond formation, alongside the optimized structures of the C–H activation transition states TS1 and TS2 (selected bond distances are labelled in Å). Energies are relative to the complex IM1 and are mass-balanced.     

An Experimental Study of a Flux Modulated Solid State Metathesis Reaction

A solid state metathesis (SSM) reaction was investigated with respect to the formation of rare‐earth carbodiimides, the role of the co‐produced salt (LiCl), and the eutectic flux medium (LiCl/KCl). A SSM reaction is characterized by an exothermic reaction in which a salt (often LiCl) is coproduced. When the salt melts, it can serve as a useful medium for the crystallization of a desired product. An improved crystal growth can be observed by using an eutectic flux. However, the composition of an eutectic LiCl/KCl flux is altered when LiCl is produced during the reaction. The thermal effects concerning the endothermic melting of the flux and the exothermic ingnition of the SSM reaction may compensate each other, which is not necessarily a drawback for the reaction to proceed.     

Magnetic Anisotropy of Dichlorobis(η5‐cyclopentadienyl) Complexes of Vanadium,Niobium, and Tantalum

Abstract. The magnetic behavior of the mononuclear nd¹ systems MCp₂Cl₂ (M = V⁴⁺[3d¹], Nb⁴⁺[4d¹], Ta⁴⁺[5d¹], space group P2₁/c, pseudosymmetry of the molecules C_2v) deviates from pure single ion spin magnetism on account of ligand field effect (H_lf), spin‐orbit coupling (H_so), and intermolecular spin‐spin exchange interactions (H_ex). For both VCp₂Cl₂ and NbCp₂Cl₂ excellent adaptations to the measured susceptibility data were obtained (2 K ≤ T ≤ 300 K) on the basis of spectroscopic data (lf, so) and cooperative metal–metal interactions (ex) of antiferromagnetic nature [molecular field model (mf)]. For TaCp₂Cl₂ experimental term structure data are not available. Therefore, Jørgensen's spectroscopical series (g‐factor of the central ion) was applied to extrapolate the data set for TaCp₂Cl₂. H_lf, H_so, and H_ex (antiferromagnetic) increase in the order 3d¹ → 4d¹ → 5d¹ leading, with rising atomic number of the metals, to a distinct enhancement of the magnetic anisotropy. At 4 K the μ_eff components μ_eff,y (oriented perpendicular to the cg–M–cg plane; “cg” = center of gravity of the Cp ring), μ_eff,z (oriented along the twofold pseudoaxis), and μ_eff,x are 1.73, 1.69, 1.68 (V), 1.73, 1.62, 1.59 (Nb), and 1.71, 1.59, 1.49 (Ta). While μ_eff,y is independent of T, both μ_eff,z and μ_eff,x decrease with decreasing T.