General Theorem Research Articles

On the equivalence between a conjecture of Babai-Godsil and a conjecture of Xu concerning the enumeration of Cayley graphs

In this paper we show that two distinct conjectures, the first proposed by Babai and Godsil in 1982 and the second proposed by Xu in 1998, concerning the asymptotic enumeration of Cayley graphs are in fact equivalent. This result follows from a more general theorem concerning the asymptotic enumeration of a certain family of Cayley graphs.

On the resolution of singularities of one-dimensional foliations on three-manifolds

This paper is devoted to the resolution of singularities of holomorphic vector fields and one-dimensional holomorphic foliations in dimension three, and it has two main objectives. First, within the general framework of one-dimensional foliations, we build upon and essentially complete the work of Cano, Roche, and Spivakovsky (2014). As a consequence, we obtain a general resolution theorem comparable to the resolution theorem of McQuillan–Panazzolo (2013) but proved by means of rather different methods. The other objective of this paper is to consider a special class of singularities of foliations containing, in particular, all the singularities of complete holomorphic vector fields on complex manifolds of dimension three. We then prove that a much sharper resolution theorem holds for this class of holomorphic foliations. This second result was the initial motivation for this paper. It relies on combining earlier resolution theorems for (general) foliations with some classical material on asymptotic expansions for solutions of differential equations. Bibliography: 34 titles.

Mathematical model and analysis method for flowfield separation and transition

Transition and separation are difficult but important problems in the field of fluid mechanics. Hitherto, separation and transition problems have not been described accurately in mathematical terms, leading to design errors and prediction problems in fluid machine engineering. The nonlinear uncertainty involved in separation and transition makes it difficult to accurately analyze these phenomena using experimental methods. Thus, new ideas and methods are required for the mathematical prediction of fluid separation and transition. In this article, after an axiomatic treatment of fluid mechanics, the concept of an excited state is derived by generating a fluctuation velocity, and it is revealed that fluid separation and transition are special forms of this excited state. This allows us to clarify the state conditions of fluid separation and transition. Mathematical analysis of the Navier–Stokes equations leads to a general excited state theorem suitable for flowfields. Finally, the conditions of separation and transition are derived, and the corresponding general laws are established. The results presented in this article provide a foundation for future research on the mechanism of turbulence and the solution of engineering problems.

Quasiflats in hierarchically hyperbolic spaces

The rank of a hierarchically hyperbolic space is the maximal number of unbounded factors in a standard product region. For hierarchically hyperbolic groups, this coincides with the maximal dimension of a quasiflat. Several noteworthy examples for which the rank coincides with familiar quantities include: the dimension of maximal Dehn twist flats for mapping class groups; the maximal rank of a free abelian subgroup for right-angled Coxeter groups and right-angled Artin groups (in the latter this can also be observed as the clique number of the defining graph); and, for the Weil–Petersson metric, the rank is the integer part of half the complex dimension of Teichmüller space. We prove that, in a hierarchically hyperbolic space (HHS), any quasiflat of dimension equal to the rank lies within finite distance of a union of standard orthants (under a very mild condition on the HHS satisfied by all natural examples). This resolves outstanding conjectures when applied to a number of different groups and spaces. In the case of the mapping class group, we verify a conjecture of Farb. For Teichmüller space we answer a question of Brock. In the context of certain CAT(0) cubical groups, our result handles novel special cases, including right-angled Coxeter groups. An important ingredient in the proof, which we expect will have other applications, is that the hull of any finite set in an HHS is quasi-isometric to a CAT(0) cube complex of dimension bounded by the rank. (If the HHS is a CAT(0) cube complex, then the rank can be lower than the dimension of the space.) We deduce a number of applications of these results. For instance, we show that any quasi-isometry between HHSs induces a quasi-isometry between certain factored spaces, which are simpler HHSs. This allows one, for example, to distinguish quasi-isometry classes of right-angled Artin/Coxeter groups. Another application of our results is to quasi-isometric rigidity. Our tools in many cases allow one to reduce the problem of quasi-isometric rigidity for a given hierarchically hyperbolic group to a combinatorial problem. As a template, we give a new proof of quasi-isometric rigidity of mapping class groups, which, once we have established our general quasiflats theorem, uses simpler combinatorial arguments than in previous proofs.

Minimum 0-extension problems on directed metrics

For a metric μ on a finite set T, the minimum 0-extension problem 0-Ext[μ] is defined as follows: Given V⊇T and c:V2→Q+, minimize ∑c(xy)μ(γ(x),γ(y)) subject to γ:V→T,γ(t)=t(∀t∈T), where the sum is taken over all unordered pairs in V. This problem generalizes several classical combinatorial optimization problems such as the minimum cut problem or the multiterminal cut problem. Karzanov and Hirai established a complete classification of metrics μ for which 0-Ext[μ] is polynomial time solvable or NP-hard. This result can also be viewed as a sharpening of the general dichotomy theorem for finite-valued CSPs (Thapper and Živný 2016) specialized to 0-Ext[μ].In this paper, we consider a directed version 0⃗-Ext[μ] of the minimum 0-extension problem, where μ and c are not assumed to be symmetric. We extend the NP-hardness condition of 0-Ext[μ] to 0⃗-Ext[μ]: If μ cannot be represented as the shortest path metric of an orientable modular graph with an orbit-invariant “directed” edge-length, then 0⃗-Ext[μ] is NP-hard. We also show a partial converse: If μ is a directed metric of a modular lattice with an orbit-invariant directed edge-length, then 0⃗-Ext[μ] is tractable. We further provide a new NP-hardness condition characteristic of 0⃗-Ext[μ], and establish a dichotomy for the case where μ is a directed metric of a star.

Stein’s method for the Poisson–Dirichlet distribution and the Ewens sampling formula, with applications to Wright–Fisher models

We provide a general theorem bounding the error in the approximation of a random measure of interest—for example, the empirical population measure of types in a Wright–Fisher model—and a Dirichlet process, which is a measure having Poisson–Dirichlet distributed atoms with i.i.d. labels from a diffuse distribution. The implicit metric of the approximation theorem captures the sizes and locations of the masses, and so also yields bounds on the approximation between the masses of the measure of interest and the Poisson–Dirichlet distribution. We apply the result to bound the error in the approximation of the stationary distribution of types in the finite Wright–Fisher model with infinite-alleles mutation structure (not necessarily parent independent) by the Poisson–Dirichlet distribution. An important consequence of our result is an explicit upper bound on the total variation distance between the random partition generated by sampling from a finite Wright–Fisher stationary distribution, and the Ewens sampling formula. The bound is small if the sample size n is much smaller than N1/6log(N)−1/2, where N is the total population size. Our analysis requires a result of separate interest, giving an explicit bound on the second moment of the number of types of a finite Wright–Fisher stationary distribution. The general approximation result follows from a new development of Stein’s method for the Dirichlet process, which follows by viewing the Dirichlet process as the stationary distribution of a Fleming–Viot process, and then applying Barbour’s generator approach.

The rise and fall of L-spaces, II

In 2005, Ben-El-Mechaiekh, Chebbi, and Florenzano obtained a generalization of Ky Fan's 1984 KKM theorem on the intersection of a
 family of closed sets on non-compact convex sets in a topological vector space. They also extended the Fan-Browder fixed point theorem
 to multimaps on non-compact convex sets. Since then several groups of the L-space theorists introduced coercivity families and applied
 them to L-spaces, H-spaces, etc. In this article, we show that better forms of such works can be deduced from a general KKM theorem
 on abstract convex spaces in our previous works. Consequently, all of the known KKM theoretic results on L-spaces related coercivity
 families are extended to corresponding better forms on abstract convex spaces.
 
 This article is a continuation of our \cite{38} and a revised and extended version of \cite{34}.

Tree limits and limits of random trees

AbstractWe explore the tree limits recently defined by Elek and Tardos. In particular, we find tree limits for many classes of random trees. We give general theorems for three classes of conditional Galton–Watson trees and simply generated trees, for split trees and generalized split trees (as defined here), and for trees defined by a continuous-time branching process. These general results include, for example, random labelled trees, ordered trees, random recursive trees, preferential attachment trees, and binary search trees.

Utility maximization in a multidimensional semimartingale model with nonlinear wealth dynamics

We explore martingale and convex duality techniques to maximize expected risk-averse utility from consumption in a general multi-dimensional (non-Markovian) semimartingale market model with jumps and non-linear wealth dynamics. The model allows to incorporate additional cash flows via non-linear margin payment functions in the drift term that depend on the allocation proportion. These can be used to cast frictions such as the impact of the portfolio choices of a ‘large’ investor on the expected assets’ returns, funding costs arising from differential borrowing and lending rates, and the cash inflow of a firm in a neoclassical economy with constant return-to-scale Cobb–Douglas technology subject to exogenous aggregate shocks. We provide a general verification theorem for random utility fields satisfying the usual Inada conditions, find conditions under which jumps in our model lead to precautionary saving, and present an explicit characterization for CRRA. We report two-fund separation-type results which assert that optimal allocations move along one-dimensional segments, and illustrate our results numerically for various margin payment functions and bounded variation tempered $$\alpha $$ -stable jumps.

Holomorphic maps into Grassmann manifolds (harmonic maps into Grassmann manifolds III)

A well-known Calabi’s rigidity theorem on holomorphic isometric immersions into the complex projective space is generalized to the case that the target is the complex Grassmann manifolds. Our strategy is to use the differential geometry of vector bundles and a generalization of do Carmo and Wallach theory developed in Nagatomo (Harmonic maps into Grassmann manifolds. arXiv:mathDG/1408.1504 ). We introduce the associated maps with holomorphic maps to obtain a general rigidity theorem (Theorem 5.6). As applications, several rigidity results on Einstein–Hermitian holomorphic maps are exhibited and we also give an interpretation of the existence of a Kahler structure with an $$S^1$$ -action on the moduli spaces of holomorphic isometric embeddings of a compact Kahler manifold into complex quadrics. Theorem 5.6 also implies classification theorems for equivariant holomorphic maps.

The Expected Number of Viable Autocatalytic Sets in Chemical Reaction Systems.

The emergence of self-sustaining autocatalytic networks in chemical reaction systems has been studied as a possible mechanism for modeling how living systems first arose. It has been known for several decades that such networks will form within systems of polymers (under cleavage and ligation reactions) under a simple process of random catalysis, and this process has since been mathematically analyzed. In this paper, we provide an exact expression for the expected number of self-sustaining autocatalytic networks that will form in a general chemical reaction system, and the expected number of these networks that will also be uninhibited (by some molecule produced by the system). Using these equations, we are able to describe the patterns of catalysis and inhibition that maximize or minimize the expected number of such networks. We apply our results to derive a general theorem concerning the trade-off between catalysis and inhibition, and to provide some insight into the extent to which the expected number of self-sustaining autocatalytic networks coincides with the probability that at least one such system is present.

A General Theory of Comparison of Quantum Channels (and Beyond)

We present a general theory of comparison of quantum channels, concerning with the question of simulability or approximate simulability of a given quantum channel by allowed transformations of another given channel. We introduce a modification of conditional min-entropies, with respect to the set F of allowed transformations, and show that under some conditions on F, these quantities characterize approximate simulability. If F is the set of free superchannels in a quantum resource theory of processes, the modified conditional min-entropies form a complete set of resource monotones. If the transformations in F consist of a preprocessing and a postprocessing of specified forms, approximate simulability is also characterized in terms of success probabilities in certain guessing games, where a preprocessing of a given form can be chosen and the measurements are restricted. These results are applied to several specific cases of simulability of quantum channels, including postprocessings, preprocessings and processing of bipartite channels by LOCC superchannels and by partial superchannels, as well as simulability of sets of quantum measurements. These questions are first studied in a general setting that is an extension of the framework of general probabilistic theories (GPT), suitable for dealing with channels. Here we prove a general theorem that shows that approximate simulability can be characterized by comparing outcome probabilities in certain tests. This result is inspired by the classical Le Cam randomization criterion for statistical experiments and contains its finite dimensional version as a special case.

Adaptive output feedback regulation for a class of uncertain feedforward stochastic nonlinear systems

This paper addresses the global output feedback regulation problem for a class of uncertain feedforward stochastic nonlinear systems. Unlike the previous works, the drift’s growth rate depends not only on an unknown constant but also on an output polynomial function and an input function. Moreover, the diffusion’s growth rate depends on an unknown constant or an output polynomial function and an input function. By using the general stochastic convergence theorem, an output controller is constructed to guarantee the boundedness and stochastic convergence of the resulting closed-loop system.

A new general class of RC association models: Estimation and main properties

The paper introduces a new class of row-column (RC) association models for contingency tables by allowing the user to select both the scale on which interactions are measured as in Kateri and Papaioannou (1994) and the type of logit (local, global, continuation) suitable for the row and column classification variables as in Bartolucci and Forcina (2002). These choices determine the matrix of interactions which is subjected to rank constraints. Examples are provided where the new models fit substantially better than traditional ones; an intuitive explanation for this behavior is outlined and supported by numerical investigations. An extension of the optimality property in Kateri and Papaioannou (1994) is derived, leading to a general representation theorem and reconstruction formulas for the joint probabilities. These results are the key to show that, given marginal logits, the generalized interactions introduced in this paper determine uniquely the bivariate distribution; in addition, for each pair of logit types, we establish which kind of positive association is implied by our extended interactions being non negative. Quick model selection within this wide class can be performed by an efficient algorithm for computing maximum likelihood estimates which allows for additional linear constraints both on marginal logits and interactions.

Three transport models for charged particles in three-dimensional semiconductors driven by a fractional noise

In this work, three models of the motion of charged particles in three-dimensional semiconductors governed by a stochastic differential equation driven by a magnetic field and an intrinsic fractional Gaussian noise have been introduced. Based on the general expansion theorem for the Laplace transform, the expressions of the average position and the average velocity of a charged particle have been obtained. The expressions of the complex susceptibilities, the spectral amplification, the stationary form of current density, as well as power absorption also have been obtained. It is worthy to note that the cyclotron resonance, stochastic dynamics of a charged particle could be induced by fractional noise. Furthermore, the expressions of variances and the generalized Fokker–Planck equation (GFPE) for the non-Markovian dynamics also have been investigated.

Extensive Study of the Quality of Fission Yields from Experiment, Evaluation and GEF for Antineutrino Studies and Applications

The understanding of the antineutrino production in fission and the theoretical calculation of the antineutrino energy spectra in different, also future, types of fission reactors rely on the application of the summation method, where the individual contributions from the different radioactive nuclides that undergo a beta decay are estimated and summed up. The most accurate estimation of the independent fission-product yields is essential to this calculation. This is a complex task because the yields depend on the fissioning nucleus and on the energy spectrum of the incident neutrons.In the present contribution, the quality of different sources of information on the fission yields is investigated, and the benefit of a combined analysis is demonstrated. The influence on antineutrino predictions is discussed.In a systematic comparison, the quality of fission-product yields emerging from different experimental techniques is analyzed. The traditional radiochemical method, which is almost exclusively used for evaluations, provides an unambiguous identification in Z and A, but it is restricted to a limited number of suitable targets, is slow, and the accuracy suffers from uncertainties in the spectroscopic nuclear properties. Experiments with powerful spectrometers, for example at LOHENGRIN, provide very accurate mass yields and a Z resolution for light fission products from thermal-neutron-induced fission of a few suitable target nuclei.On the theoretical side, the general fission model GEF has been developed. It combines a few general theorems, rules and ideas with empirical knowledge. GEF covers almost all fission observables and is able to reproduce measured data with high accuracy while having remarkable predictive power by establishing and exploiting unexpected systematics and hidden regularities in the fission observables. In this article, we have coupled for the first time the GEF predictions for the fission yields to fission-product beta-decay data in a summation calculation of reactor antineutrino energy spectra. The first comparisons performed between the spectra from GEF and those obtained with the evaluated nuclear databases exhibited large discrepancies that highlighted the exigency of the modelisation of the antineutrino spectra and showing their usefulness in the evaluation of nuclear data. Additional constraints for the GEF model were thus needed in order to reach the level of accuracy required by the antineutrino energy spectra. The combination of a careful study of the independent isotopic yields and the adjunction of the LOHENGRIN fission-yield data as additional constraints led to a substantially improved agreement between the antineutrino spectra computed with GEF and with the evaluated data. The comparison of inverse beta-decay yields computed with GEF with those measured by the Daya Bay experiment shows the excellent level of predictiveness of the GEF model for the fundamental or applied antineutrino physics.The main results of this study are:–an improved agreement between the antineutrino energy spectra obtained with the newly tuned GEF model and the JEFF-3.1.1 and JEFF-3.3 fission yields for the four main contributors to fission in standard power reactors;–indications for shortcomings of mass yields for 241Pu(nth, f) and other systems in current evaluations;–a demonstration of the benefit from cross-checking the results of different experimental approaches and GEF for improving the quality of nuclear data;–an analysis of the sources of uncertainties and erroneous results from different experimental approaches;–the capacity of GEF for predicting the fission yields (and other observables) in cases (in terms of fissioning systems and excitation energies) which are presently not accessible to experiment;–predictions of antineutrino energy spectra that aim to assess the prospects for reactor monitoring, and based on the GEF fission yields associated with the beta-decay data of the most recent summation model.

Fundamental Limits of Lossless Data Compression With Side Information

The problem of lossless data compression with side information available to both the encoder and the decoder is considered. The finite-blocklength fundamental limits of the best achievable performance are defined, in two different versions of the problem: Reference-based compression, when a single side information string is used repeatedly in compressing different source messages, and pair-based compression, where a different side information string is used for each source message. General achievability and converse theorems are established for arbitrary source-side information pairs. Nonasymptotic normal approximation expansions are proved for the optimal rate in both the reference-based and pair-based settings, for memoryless sources. These are stated in terms of explicit, finite-blocklength bounds, that are tight up to third-order terms. Extensions that go significantly beyond the class of memoryless sources are obtained. The relevant source dispersion is identified and its relationship with the conditional varentropy rate is established. Interestingly, the dispersion is different in reference-based and pair-based compression, and it is proved that the reference-based dispersion is in general smaller.

A Pedagogical Model of General Covariance Coupling Electrodynamics and Gravitation

In his 1951 model of the effects of a star’s radiation on its gravitation, P.C. Vaidya used radiation fields of flat spacetime. Wondering why Vaidya did not include the effects of gravitation on electrodynamics, we apply the principle of general covariance to gravitation and radiation. Others have addressed this problem with numerical methods, but, like Vaidya, we seek an analytic solution. Along the way, we see how prescient was Vaidya’s choice in modeling the radiation. Pursued originally as a pedagogical exercise with undergraduate physicists, our results compare favorably to similar models, and illustrate general relativity theorems. Given recent interest in teaching general relativity to undergraduates, a system that couples gravitation and electrodynamics and is solvable with elementary integrations may hold pedagogical interest.

A note on simultaneous approximation on Vitushkin sets

Given a planar Jordan domain $G$ with rectifiable boundary, it is well known that smooth functions on the closure of $G$ do not always admit smooth extensions to $\mathbb C$. Further conditions on the boundary are necessary to guarantee such extensions. On the other hand, Weierstrass’ approximation theorem yields polynomials converging uniformly to $f \in C(\overline{G}, \mathbb C)$. In this note we show that for Vitushkin sets $K$ with $K = \overline{K^\circ}$ it is always possible to uniformly approximate on $K$ the smooth function $f \in C^1(K, \mathbb C)$ by smooth functions $f_n$ in $\mathbb C$ so that also $\overline{\partial}f_n$ converges uniformly to $\overline{\partial}f$ on $K$. As a byproduct we deduce from its ‘‘smooth in a neighborhood version’’ the general Gauss integral theorem for functions whose partial derivatives in $G$ merely admit continuous extensions to its boundary.

Two Classes of Iteration Functions and Q-Convergence of Two Iterative Methods for Polynomial Zeros

In this work, two broad classes of iteration functions in n-dimensional vector spaces are introduced. They are called iteration functions of the first and second kind at a fixed point of the corresponding iteration function. Two general local convergence theorems are presented for Picard-type iterative methods with high Q-order of convergence. In particular, it is shown that if an iterative method is generated by an iteration function of first or second kind, then it is Q-convergent under each initial approximation that is sufficiently close to the fixed point. As an application, a detailed local convergence analysis of two fourth-order iterative methods is provided for finding all zeros of a polynomial simultaneously. The new results improve the previous ones for these methods in several directions.