Absolute Constants Research Articles

A minimax framework for quantifying risk-fairness trade-off in regression

We propose a theoretical framework for the problem of learning a real-valued function which meets fairness requirements. This framework is built upon the notion of α-relative (fairness) improvement of the regression function which we introduce using the theory of optimal transport. Setting α=0 corresponds to the regression problem under the Demographic Parity constraint, while α=1 corresponds to the classical regression problem without any constraints. For α∈(0,1) the proposed framework allows to continuously interpolate between these two extreme cases and to study partially fair predictors. Within this framework, we precisely quantify the cost in risk induced by the introduction of the fairness constraint. We put forward a statistical minimax setup and derive a general problem-dependent lower bound on the risk of any estimator satisfying α-relative improvement constraint. We illustrate our framework on a model of linear regression with Gaussian design and systematic group-dependent bias, deriving matching (up to absolute constants) upper and lower bounds on the minimax risk under the introduced constraint. We provide a general post-processing strategy which enjoys fairness, risk guarantees and can be applied on top of any black-box algorithm. Finally, we perform a simulation study of the linear model and numerical experiments of benchmark data, validating our theoretical contributions.

Combined Experimental and Computational Kinetics Studies for the Atmospherically Important BrHg Radical Reacting with NO and O2.

We report on the first experimental determination of the absolute rate constant of the reaction of BrHg + NO in N2 bath gas using a laser photolysis-laser-induced fluorescence (LP-LIF) system. The rate constant of the reaction of BrHg + NO is determined to be 7.0-0.9+1.3 × 10-12 cm3 molecule-1 s-1 over 50-700 Torr and 315-353 K. The absence of a pressure or temperature dependence suggests that this reaction leads mainly to mercury reduction (Hg + BrNO) rather than mercury oxidation (BrHgNO). Our theoretical calculations using NEVPT2 energies on density functional theory (DFT) geometries are consistent with a barrierless reaction to form Hg + BrNO. The equilibrium constants and the rate constants of the reaction BrHg + O2 ⇌ BrHgOO are computed theoretically because they are too low to be measured in the LP-LIF system. Molecular oxygen quenches the LIF signal of BrHg with a large rate constant of (1.7 ± 0.1) × 10-10 cm3 molecule-1 s-1. Thus, different techniques that capture the absolute [BrHg(X̃)] would be advantageous for kinetics studies of BrHg reactions in the presence of O2. The computed equilibrium constant suggests a non-negligible upper limit of the fraction of BrHg stored as BrHgOO (up to 0.5) at low-temperature conditions, e.g., in the upper troposphere and in polar winters at ground level. Preliminary results indicate that BrHgOO behaves like HOO or organic peroxy radicals in reactions with atmospheric radicals.

Theoretical Study of the Isotope and Homologue Effects on Nuclear Magnetic Shielding in Water and Hydrogen Sulfide Molecules

Abstract The hydrogen/deuterium isotope and oxygen/sulfur homologue effects on the absolute nuclear magnetic shielding constants were theoretically analyzed for five 17O-water isotopologues and five hydrogen sulfide isotopologues. Using both ab initio calculations and anharmonic vibrational structure calculations based on vibrational quantum Monte Carlo theory, we confirmed that the changes in the atomic charges are strongly correlated with the enhancements of the absolute nuclear magnetic shielding constants.

Hitting Time of Edge Disjoint Hamilton Cycles in Random Subgraph Processes on Dense Base Graphs

Consider the random subgraph process on a base graph $G$ on $n$ vertices: a sequence $\lbrace G_t \rbrace _{t=0} ^{|E(G)|}$ of random subgraphs of $G$ obtained by choosing an ordering of the edges of $G$ uniformly at random, and by sequentially adding edges to $G_0$, the empty graph on the vertex set of $G$, according to the chosen ordering. We show that if $G$ has one of the following properties: 1. There is a positive constant $\varepsilon > 0$ such that $\delta (G) \geq \left( \frac{1}{2} + \varepsilon \right) n$; 2. There are some constants $\alpha, \beta >0$ such that every two disjoint subsets $U,W$ of size at least $\alpha n$ have at least $\beta |U||W|$ edges between them, and the minimum degree of $G$ is at least $(2\alpha + \beta )\cdot n$; or: 3. $G$ is an $(n,d,\lambda )$--graph, with $d\geq \frac{C\cdot n\cdot \log \log n}{\log n}$ and $\lambda \leq \frac{c\cdot d^2}{n}$ for some absolute constants $c,C>0$. then for a positive integer constant $k$ with high probability the hitting time of the property of containing $k$ edge disjoint Hamilton cycles is equal to the hitting time of having minimum degree at least $2k$. These results extend prior results by by Johansson and by Frieze and Krivelevich, and answer a question posed by Frieze.

Cryo kinetics of N2 adsorption onto bimetallic rhodium-iron clusters in isolation.

We report the N2 cryo adsorption kinetics of selected gas phase mixed rhodium-iron clusters [RhiFej]+ in the range of i = 3-8 and j = 3-8 in 26K He buffer gas by the use of a cryo tandem RF-hexapole trap-Fourier transform ion cyclotron resonance mass spectrometer. From kinetic data and fits, we extract relative rate constants for each N2 adsorption step and possible desorption steps. We find significant trends in adsorption behavior, which reveal adsorption limits, intermittent adsorption limits, and equilibrium reactions. For those steps, which are in equilibrium, we determine the Gibbs free energies. We conclude on likely ligand shell reorganization and some weakly bound N2 ligands for clusters where multiple N2 adsorbates are in equilibrium. The relative rate constants are transferred to absolute rate constants, which are slightly smaller than the collision rate constants calculated by the average dipole orientation (Langevin) theory. The calculated sticking probabilities increase, in general, with the size of the clusters and decrease with the level of N2 adsorption, in particular, when reaching an adsorption/desorption equilibrium. We receive further evidence on cluster size dependent properties, such as cluster geometries and metal atom distributions within the clusters through the accompanying spectroscopic and computational study on the equiatomic i = j clusters [Klein et al., J. Chem. Phys. 156, 014302 (2022)].

Extended Benchmark Set of Main-Group Nuclear Shielding Constants and NMR Chemical Shifts and Its Use to Evaluate Modern DFT Methods.

An extended theoretical benchmark set, NS372, for light main-group nuclear shieldings and NMR shifts has been constructed based on high-level GIAO-CCSD(T)/pcSseg-3//CCSD(T)/cc-pVQZ reference data. After removal of the large static-correlation cases O3, F3-, and BH from the statistical evaluations for the 17O, 19F, and 11B subsets, the benchmark comprises overall 372 shielding values in 117 molecules with a wide range of electronic-structure situations, containing 124 1H, 14 11B, 93 13C, 43 15N, 31 17O, 47 19F, 14 31P, and 6 33S shielding constants. The CCSD(T)/pcSseg-3 data are shown to be close to the basis-set and method limit and thus provide an excellent benchmark to evaluate more approximate methods, such as density functional approaches. This dataset has been used to evaluate Hartree-Fock (HF) and MP2, and a wide range of exchange-correlation functionals from local density approximation (LDA) to generalized gradient approximations (GGAs) and meta-GGAs (focusing on their current-density functional implementations), as well as global hybrid, range-separated hybrid, local hybrid, and double-hybrid functionals. Starting with absolute shielding constants, the DSD-PBEP86 double hybrid is confirmed to provide the highest accuracy, with an aggregate relative mean absolute error (rel. MAE) of only 0.9%, followed by MP2 (1.1%). MP2 and double hybrids only show larger errors for a few systems with the largest static-correlation effects. The double-hybrid B2GP-PLYP, the two local hybrids cLH12ct-SsirPW92 and cLH12ct-SsifPW92, and the current-density functional meta-GGA cB97M-V follow closely behind (all 1.5%), as do some further functionals, cLH20t and cMN15-L (both 1.6%), as well as B2PLYP and KT3 (both 2.0%). Functionals on the lower rungs of the usual ladder offer the advantage of lower computational cost and access to larger molecules. Closer examination also reveals the best-performing methods for individual nuclei in the test set. Different ways of treating τ-dependent functionals are evaluated. When moving from absolute shielding constants to chemical shifts, some of the methods can benefit from systematic error compensation, and the overall error range somewhat narrows. Further methods now achieve the 2% threshold of relative MAEs, including functionals based on TPSS (TPSSh, cmPSTS).

Charge-Separation and Charge-Recombination Rate Constants in a Donor-Acceptor Buckybowl-Based Supramolecular Complex: Multistate and Solvent Effects.

The kinetics of the nonradiative photoinduced processes (charge-separation and charge-recombination) experimented in solution by a supramolecular complex formed by an electron-donating bowl-shaped truxene-tetrathiafulvalene (truxTTF) derivative and an electron-accepting fullerene fragment (hemifullerene, C30H12) has been theoretically investigated. The truxTTF·C30H12 heterodimer shows a complex decay mechanism after photoexcitation with the participation of several low-lying excited states of different nature (local and charge-transfer excitations) all close in energy. In this scenario, the absolute rate constants for all of the plausible charge-separation (CS) and charge-recombination (CR) channels have been successfully estimated using the Marcus–Levich–Jortner (MLJ) rate expression, electronic structure calculations, and a multistate diabatization method. The outcomes suggest that for a reasonable estimate of the CS and CR rate constants, it is necessary to include the following: (i) optimally tuned long-range (LC) corrected density functionals, to predict a correct energy ordering of the low-lying excited states; (ii) multistate effects, to account for the electronic couplings; and (iii) environmental solvent effects, to provide a proper stabilization of the charge-transfer excited states and accurate external reorganization energies. The predicted rate constants have been incorporated in a simple but insightful kinetic model that allows estimating global CS and CR rate constants in line with the most generalized three-state model used for the CS and CR processes. The values computed for the global CS and CR rates of the donor–acceptor truxTTF·C30H12 supramolecular complex are found to be in good agreement with the experimental values.

On Calculation of Constants in the Arak Inequalities for Concentration Functions of Convolutions of Probability Distributions

The aim of the present work is evaluation of absolute constants in the Arak inequalities for the concentration functions of convolutions of probability distributions. This result allows us to calculate the constant in the inequality for the uniform distance between n and (n + 1)-fold convolutions of one-dimensional symmetric probability distributions with a characteristic function separated from −1, as well as a number of other estimates, in particular, the accuracy of approximation of samples of rare events by the Poisson point process.

Asymptotics of the Sum of a Sine Series with a Convex Slowly Varying Sequence of Coefficients

We study the asymptotic behavior in a neighborhood of zero of the sum of a sine series g(b,x)=∑k=1∞bksinkx whose coefficients constitute a convex slowly varying sequence b. The main term of the asymptotics of the sum of such a series was obtained by Aljančić, Bojanić, and Tomić. To estimate the deviation of g(b,x) from the main term of its asymptotics bm(x)/x, m(x)=[π/x], Telyakovskiĭ used the piecewise-continuous function σ(b,x)=x∑k=1m(x)−1k2(bk−bk+1). He showed that the difference g(b,x)−bm(x)/x in some neighborhood of zero admits a two-sided estimate in terms of the function σ(b,x) with absolute constants independent of b. Earlier, the author found the sharp values of these constants. In the present paper, the asymptotics of the function g(b,x) on the class of convex slowly varying sequences in the regular case is obtained.

Role of pH in the Transformation of Perfluoroalkyl Carboxylic Acids by Activated Persulfate: Implications from the Determination of Absolute Electron-Transfer Rates and Chemical Computations.

Perfluoroalkyl carboxylic acids (PFCAs) are ubiquitous contaminants known for their bioaccumulation, toxicological harm, and resistance to degradation. Remediating PFCAs in water is an ongoing challenge with existing technologies being insufficient or requiring additional disposal. An emergent approach is using activated persulfate, which degrades PFCAs through sequential scission of CF2 equivalents yielding shorter-chain homologues, CO2 and F-. This transformation is thought to be initiated by single electron transfer (SET) from the PFCA to the activate oxidant, SO4•-. A pronounced pH effect has been observed for thermally activated persulfate PFCA transformation. To evaluate the role of pH during SET, we directly determined absolute rate constants for perfluorobutanoic acid and trifluoroacetic acid oxidation by SO4•- in the pH range of 0.5-4.0 using laser flash photolysis. The average of the rate constants for both substrates across all pH values was 9 ± 2 × 103 M-1 s-1 (±2σ), implying that acid catalysis of thermal persulfate activation may be the primary culprit of the observed pH effect, instead of pH influencing the SET step. In addition, density functional theory was used to investigate if SO4•-protonation might enhance PFCA transformation kinetics. We found that when calculations include explicit water molecules, direct SO4•- protonation does not occur.

Insight into Hydrogen Abstractions by Nitrate Radical: Structural, Solvent Effects, and Evidence for a Polar Transition State.

The role of a polarized transition state and solvent effects on nitrate radical reactions was examined with a previously under-reported class of substrates, ethers, for their atmospheric implications. Absolute rate constants for hydrogen abstraction from a series of alcohols, ethers, and alkanes by nitrate radical have been measured in acetonitrile, water, and mixtures of these two solvents. Across all of these classes of compounds, using a modified form of the Evans-Polanyi relationship, it is demonstrated that the observed structure/reactivity trends can be reconciled by considering the number of abstractable hydrogens, strength of the C-H bond, and ionization potential (IP) of the substrate. Hydrogen abstractions by nitrate radical occur with low selectivity and are characterized by an early transition state (α ≈ 0.3). The dependence of the rate constant on IP suggests a polar transition state with some degree (<10%) of charge transfer. These conclusions stand for reactions conducted in solution (CH3CN and H2O) as well as gas-phase values. Because of this polar transition state, the rate constants increase going from the gas phase to a polar solvent, and the magnitude of the increase is consistent with Kirkwood theory.

Actions of the additive group Ga on certain noncommutative deformations of the plane

Abstract We connect the theorems of Rentschler [18] and Dixmier [10] on locally nilpotent derivations and automorphisms of the polynomial ring A 0 and of the Weyl algebra A 1, both over a field of characteristic zero, by establishing the same type of results for the family of algebras A h = 〈 x , y | y x − x y = h ( x ) 〉 , {A_h} = \left\langle {x,y|yx - xy = h\left( x \right)} \right\rangle , , where h is an arbitrary polynomial in x. In the second part of the paper we consider a field 𝔽 of prime characteristic and study 𝔽[t]-comodule algebra structures on Ah . We also compute the Makar-Limanov invariant of absolute constants of Ah over a field of arbitrary characteristic and show how this subalgebra determines the automorphism group of Ah .

Redox-Induced Structural Reorganization Dictates Kinetics of Cobalt(III) Hydride Formation via Proton-Coupled Electron Transfer.

Two-electron, one-proton reactions of a family of [CoCp(dxpe)(NCCH3)]2+ complexes (Cp = cyclopentadienyl, dxpe = 1,2-bis(di(aryl/alkyl)phosphino)ethane) form the corresponding hydride species [HCoCp(dxpe)]+ (dxpe = dppe (1,2-bis(diphenylphosphino)ethane), depe (1,2-bis(diethylphosphino)ethane), and dcpe (1,2-bis(dicyclohexylphosphino)ethane)) through a stepwise proton-coupled electron transfer process. For three [CoCp(dxpe)(NCCH3)]2+ complexes, peak shift analysis was employed to quantify apparent proton transfer rate constants from cyclic voltammograms recorded with acids ranging 22 pKa units. The apparent proton transfer rate constants correlate with the strength of the proton source for weak acids, but these apparent proton transfer rate constants curiously plateau (kpl) as the reaction becomes increasingly exergonic. The absolute apparent proton transfer rate constants across both these regions correlate with the steric bulk of the chelating diphosphine ligand, with bulkier ligands leading to slower kinetics (kplateau,depe = 3.5 × 107 M-1 s-1, kplateau,dppe = 1.7 × 107 M-1 s-1, kplateau,dcpe = 7.1 × 104 M-1 s-1). Mechanistic studies were conducted to identify the cause of the aberrant kPTapp-ΔpKa trends. When deuterated acids are employed, deuterium incorporation in the Cp ring is observed, indicating protonation of the CoCp(dxpe) species to form the corresponding hydride proceeds via initial ligand protonation. Digital simulations of cyclic voltammograms show ligand loss accompanying initial reduction gates subsequent PCET activity at higher driving forces. Together, these experiments reveal the details of the reaction mechanism: reduction of the Co(III) species is followed by dissociation of the bound acetonitrile ligand, subsequent reduction of the unligated Co(II) species to form a Co(I) species is followed by protonation, which occurs at the Cp ring, followed by tautomerization to generate the stable Co(III)-hydride product [HCoCp(dxpe)]+. Analysis as a function of chelating disphosphine ligand, solvent, and acid strength reveals that the ligand dissociation equilibrium is directly influenced by the steric bulk of the phosphine ligands and gates protonation, giving rise to the plateau of the apparent proton transfer rate constant with strong acids. The complexity of the reaction mechanism underpinning hydride formation, encompassing dynamic behavior of the entire ligand set, highlights the critical need to understand elementary reaction steps in proton-coupled electron transfer reactions.

First-principles study of the atomic volume of hydrogen in palladium

The partial atomic volume of hydrogen, vH, is a fundamentally important thermodynamic parameter of interstitial metal hydrides in which dissociated H occupies interstices in the metal lattice. Such an important property should be able to be reliably calculated by a suitable theory or model in order to explain and understand its origin. In practice, vH is typically obtained by means of ab initio calculations founded on density functional theory (DFT), where the equilibrium lattice constant at zero temperature is found by minimising the Born-Oppenheimer energy. While the absolute lattice constants calculated in this way depend quite strongly on the DFT scheme employed, the present work showed that vH is rather robust against differing calculational approaches, thus making a meaningful comparison of theory and experiment possible. Comparing vH for PdnH (0<n<8) calculated with DFT and obtained from in-situ neutron diffraction measurements revealed a significant discrepancy when octahedral-only interstitial occupancy was assumed. Calculations for PdH with mixed octahedral and tetrahedral occupancy gave a value for vH in agreement with experiment assuming that PdH contains 15–20% tetrahedral H.

Computational predictions of metal-macrocycle stability constants require accurate treatments of local solvent and pH effects.

Rational design of molecular chelating agents requires a detailed understanding of physicochemical ligand-metal interactions in solvent phase. Computational quantum chemistry methods should be able to provide this, but computational reports have shown poor accuracy when determining absolute binding constants for many chelating molecules. To understand why, we compare and benchmark static- and dynamics-based computational procedures for a range of monovalent and divalent cations binding to a conventional cryptand molecule: 2.2.2-cryptand ([2.2.2]). The benchmarking comparison shows that dynamics simulations using standard OPLS-AA classical potentials can reasonably predict binding constants for monovalent cations, but these procedures fail for divalent cations. We also consider computationally efficient static procedure using Kohn-Sham density functional theory (DFT) and cluster-continuum modeling that accounts for local microsolvation and pH effects. This approach accurately predicts binding energies for monovalent and divalent cations with an average error of 3.2 kcal mol-1 compared to experiment. This static procedure thus should be useful for future molecular screening efforts, and high absolute errors in the literature may be due to inadequate modeling of local solvent and pH effects.

A conceptual essay on the nature of spacetime

A conceptual essay on the nature of spacetime

Photoneutron production and characterization in fluoride-salt cooled high-temperature reactors

Due to the presence of beryllium-bearing salts, fluoride-salt cooled high-temperature reactors (FHRs) are expected to have higher production of delayed photoneutrons than other common power reactor systems which may impact transient behavior. New methods for computationally quantifying the characteristics of delayed photoneutrons using Monte Carlo particle transport, including absolute fraction, effective fraction, and group decay constants, are outlined. These methods are validated where possible against existing data from CANDU reactors, and then used to make a best-estimate characterization of the delayed photoneutron effect in a prototypical FHR design. It is found that delayed photoneutrons have a worth of approximately 9 pcm in the reference FHR system, and that their importance is slightly higher than that of fission neutrons due mostly to their softer spectrum. Existing emission group structures from literature have been compared to computational data and found to be lacking, particularly in their representation of groups with short half-lives. Therefore, a new 13-group structure is proposed for use in FHR dynamic calculations.

The strong circular law: A combinatorial view

Let [Formula: see text] be an [Formula: see text] complex random matrix, each of whose entries is an independent copy of a centered complex random variable [Formula: see text] with finite nonzero variance [Formula: see text]. The strong circular law, proved by Tao and Vu, states that almost surely, as [Formula: see text], the empirical spectral distribution of [Formula: see text] converges to the uniform distribution on the unit disc in [Formula: see text]. A crucial ingredient in the proof of Tao and Vu, which uses deep ideas from additive combinatorics, is controlling the lower tail of the least singular value of the random matrix [Formula: see text] (where [Formula: see text] is fixed) with failure probability that is inverse polynomial. In this paper, using a simple and novel approach (in particular, not using machinery from additive combinatorics or any net arguments), we show that for any fixed complex matrix [Formula: see text] with operator norm at most [Formula: see text] and for all [Formula: see text], [Formula: see text] where [Formula: see text] is the least singular value of [Formula: see text] and [Formula: see text] are positive absolute constants. Our result is optimal up to the constants [Formula: see text] and the inverse exponential-type error rate improves upon the inverse polynomial error rate due to Tao and Vu. Our proof relies on the solution to the so-called counting problem in inverse Littlewood–Offord theory, developed by Ferber, Luh, Samotij, and the author, a novel complex anti-concentration inequality, and a “rounding trick” based on controlling the [Formula: see text] operator norm of heavy-tailed random matrices.

Sharp Two-Sided Estimate for the Sum of a Sine Series with Convex Slowly Varying Sequence of Coefficients

The sum of a sine series $$g\left({b,x} \right) = \sum\nolimits_{k = 1}^\infty {}$$ bk sin kx with coefficients forming a convex sequence b is known to be positive on the interval (0,π). To estimate its values near zero Telyakovskiĭ used the piecewise-continuous function $$\sigma \left({{\bf{b}},x} \right) = \left({1/m\left(x \right)} \right)\sum\nolimits_{k = 1}^{m\left(x \right) - 1} {{k^2}\left({{b_k} - {b_{k + 1}}} \right),\,\,m\left(x \right) = \left[{\pi /x} \right]}$$ . He showed that in some neighborhood of zero the difference g(b,x) − (bm(x)/2) cot(x/2) can be estimated from both sides two-sided in terms of the function σ(b,x) with absolute constants. In the present paper, sharp values of these constants on the class of convex slowly varying sequences b are found. A sharp two-sided estimate for the sum of a sine series on this class is obtained. Examples that demonstrate good accuracy of the obtained two-sided estimates are given.

Standard Electrode Potentials for Electrochemical Hydrogen Production, Carbon Dioxide Reduction, and Oxygen Reduction Reactions in N,N-Dimethylacetamide

Four standard electrode potentials versus the standard hydrogen electrode (SHE) are reported in a polar aprotic solvent, N,N-dimethylacetamide (DMA) at 298 K: (1) −0.319 ± 0.033 V for a H2 production; (2) −0.415 ± 0.033 V and −1.195 ± 0.033 V for CO2 reductions to produce carbon monoxide (CO) in the absence and presence of water, respectively; and (3) +0.918 ± 0.033 V for an O2 reduction (ORR) or O2 evolution (OER) reactions. These potentials were estimated based on measurements of absolute (and conventional) acidity constants for several Brønsted acids in DMA including CO2 with water ([H2O] = 1.1 M) of which absolute pKa is 18.6 ± 0.2 in reference to aqueous proton at standard states. The standard Gibbs transfer energy of proton was calculated to be −31 ± 3 kJ/mol from water to DMA at 298 K.