arXiv subjects
Publications and source records attributed to David Goldberg.
We have computed a table of the triangle sides of all congruent numbers less than 10,000, which improves and extends the existing public table. We give some background on properties of the triangle sides, and explain how we computed our table, which is available on https://github.com/dgpaloalto/Congruent-Numbers
A/B testing is ubiquitous within the machine learning and data science operations of internet companies. Generically, the idea is to perform a statistical test of the hypothesis that a new feature is better than the existing platform---for example, it results in higher revenue. If the p value for the test is below some pre-defined threshold---often, 0.05---the new feature is implemented. The difficulty of choosing an appropriate threshold has been noted before, particularly because dependent tests are often done sequentially, leading some to propose control of the false discovery rate (FDR) rather than use of a single, universal threshold. However, it is still necessary to make an arbitrary choice of the level at which to control FDR. Here we suggest a decision-theoretic approach to determining whether to adopt a new feature, which enables automated selection of an appropriate threshold. Our method has the basic ingredients of any decision-theory problem: a loss function, action space, and a notion of optimality, for which we choose Bayes risk. However, the loss function and the action space differ from the typical choices made in the literature, which has focused on the theory of point estimation. We give some basic results for Bayes-optimal thresholding rules for the feature adoption decision, and give some examples using eBay data. The results suggest that the 0.05 p-value threshold may be too conservative in some settings, but that its widespread use may reflect an ad-hoc means of controlling multiplicity in the common case of repeatedly testing variants of an experiment when the threshold is not reached.
We study $R$-groups for $p$-adic inner forms of quasi-split special unitary groups. We prove Arthur's conjecture, the isomorphism between the Knapp-Stein $R$-group and the Langlands-Arthur $R$-group, for quasi-split special unitary groups and their inner forms. Furthermore, we investigate the invariance of the Knapp-Stein $R$-group within $L$-packets and between inner forms. This work is applied to transferring known results in the second-named author's earlier work for quasi-split special unitary groups to their non quasi-split inner forms.
In these notes we compute the Plancherel measures associated with genuine principal series representations of n-fold covers of p-adic SL(2,F). Along the way we also compute a higher dimensional metaplectic analog of Shahidi local coefficients. Our method involves new functional equations utilizing the Tate Gamma-factor and a metaplectic counterpart. As an application we prove an irreducibility theorem.
We study the reducibility of parabolically induced representations of non-split inner forms of quasi-split classical groups. The isomorphism of Arthur R-groups, endoscopic R-groups and Knapp-Stein R-groups is established, as well as showing these R-groups are isomorphic to the corresponding ones for the quasi-split form. This shows R-groups are an invariant of the L-packets. The results are applied to classify the elliptic spectrum.
We study parabolically induced representations for $GSpin_m(F)$ with $F$ a $p$--adic field of characteristic zero. The Knapp-Stein $R$--groups are described and shown to be elementary two groups. We show the associated cocycle is trivial proving multiplicity one for induced representations. We classify the elliptic tempered spectrum. For $GSpin_{2n+1}(F)$, we describe the Arthur (Endoscopic) $R$--group attached to Langlands parameters, and show these are isomorphic to the corresponding Knapp-Stein $R$--groups.
We study the Knapp-Stein $R$--groups for inner forms of the split group $SL_n(F),$ with $F$ a $p$--adic field of characteristic zero. Thus, we consider the groups $SL_m(D),$ with $D$ a central division algebra over $F$ of dimension $d^2,$ and $m=n/d.$ We use the generalized Jacquet-Langlands correspondence and results of the first named author to describe the zeros of Plancherel measures. Combined with a study of the behavior of the stabilizer of representations by elements of the Weyl group we are able to determine the Knapp-Stein $R$--groups in terms of those for $SL_n(F).$ We show the $R$--group for the inner form embeds as a subgroup of the $R$--group for the split form, and we characterize the quotient. We are further able to show the Knapp-Stein $R$--group is isomorphic to the Arthur, or Endoscopic $R$--group as predicted by Arthur. Finally, we give some results on multiplicities and actions of Weyl groups on $L$--packets.
For classical groups we show the isomorphism of the Knapp-Stein $R$-group, which describes the structure of parabolically induced representations, and the Arthur $R$-group of the parameter associated to the inducing representation by the local Langlands conjecture. We do this in the case of inducing from discrete series representations. In the case of unitary groups we show this isomorphism under a mild assumption on the parameter, which we show holds in at least half the cases.
We show that principal series representations of the real split odd orthogonal group have non-vanishing rank one bessel models of a very particular type.
We consider a decision network on an undirected graph in which each node corresponds to a decision variable, and each node and edge of the graph is associated with a reward function whose value depends only on the variables of the corresponding nodes. The goal is to construct a decision vector which maximizes the total reward. This decision problem encompasses a variety of models, including maximum-likelihood inference in graphical models (Markov Random Fields), combinatorial optimization on graphs, economic team theory and statistical physics. The network is endowed with a probabilistic structure in which costs are sampled from a distribution. Our aim is to identify sufficient conditions to guarantee average-case polynomiality of the underlying optimization problem. We construct a new decentralized algorithm called Cavity Expansion and establish its theoretical performance for a variety of models. Specifically, for certain classes of models we prove that our algorithm is able to find near optimal solutions with high probability in a decentralized way. The success of the algorithm is based on the network exhibiting a correlation decay (long-range independence) property. Our results have the following surprising implications in the area of average case complexity of algorithms. Finding the largest independent (stable) set of a graph is a well known NP-hard optimization problem for which no polynomial time approximation scheme is possible even for graphs with largest connectivity equal to three, unless P=NP. We show that the closely related maximum weighted independent set problem for the same class of graphs admits a PTAS when the weights are i.i.d. with the exponential distribution. Namely, randomization of the reward function turns an NP-hard problem into a tractable one.
We continue our study of the poles of local Langlands L-functions through the theory of induced from supercuspidal representations of quasi-split groups. Here we study the odd special orthogonal groups, and hence determine poles of Rankin product L-functions. The pole of the intertwining operator is determined in terms of the theory of orbital integrals. This gives a description of the poles in terms of twisted endoscopy, as in previous cases. We use the language of functorial transfer to give precise descrption of the pole in terms of the local components of the global transfer, which has now been established.
We derive new results for the performance of a simple greedy algorithm for finding large independent sets and matchings in constant degree regular graphs. We show that for $r$-regular graphs with $n$ nodes and girth at least $g$, the algorithm finds an independent set of expected cardinality $f(r)n - O\big(\frac{(r-1)^{\frac{g}{2}}}{\frac{g}{2}!} n\big)$, where $f(r)$ is a function which we explicitly compute. A similar result is established for matchings. Our results imply improved bounds for the size of the largest independent set in these graphs, and provide the first results of this type for matchings. As an implication we show that the greedy algorithm returns a nearly perfect matching when both the degree $r$ and girth $g$ are large. Furthermore, we show that the cardinality of independent sets and matchings produced by the greedy algorithm in \emph{arbitrary} bounded degree graphs is concentrated around the mean. Finally, we analyze the performance of the greedy algorithm for the case of random i.i.d. weighted independent sets and matchings, and obtain a remarkably simple expression for the limiting expected values produced by the algorithm. In fact, all the other results are obtained as straightforward corollaries from the results for the weighted case.
We study components of the Bernstein category for a p-adic classical group (with p odd) with inertial support a self-dual positive level supercuspidal representation of a Siegel Levi subgroup. More precisely, we use the method of covers to construct a Bushnell-Kutzko type for such a component. A detailed knowledge of the Hecke algebra of the type should have number-theoretic implications.
Let T be a complex torus, and X the surface CP^1 x T. If T is embedded in CP^{n-1} then X may be embedded in CP^{2n-1}. Let X_Gal be its Galois cover with respect to a generic projection to CP^2. In this paper we compute the fundamental group of X_Gal, using the degeneration and regeneration techniques, the Moishezon-Teicher braid monodromy algorithm and group calculations. We show that pi_1(X_Gal) = Z^{4n-2}.
We examine the reducibility of induced from discrete series representations of $SU_n$ over a $p$--adic field of characteristic zero. Some results are given for groups sharing derived group. We give a relationship between the $R$-groups for $U)n$ and $SU_n.$ Nonabelian $R$-groups are discussed. Elliptic representations are examined and in the generic case we give an explicit description f those $L$--packets containing non-discrete elliptic tempered representations. We exhibit $L$--packets containing both elliptic and non-elliptic components.
Let T be the complex projective torus, and X the surface CP^1 X T. Let X_Gal be its Galois cover with respect to a generic projection to CP^2. In this paper we compute the fundamental group of X_Gal, using the degeneration and regeneration techniques, the Moishezon-Teicher braid monodromy algorithm and group calculations. We show that pi_1(X_Gal) = Z^10.
This paper is concerned with representations of split orthogonal and quasi-split unitary groups over a nonarchimedean local field which are not generic, but which support a unique model of a different kind, the generalized Bessel model. The properties of the Bessel models under induction are studied, and an analogue of Rodier's theorem concerning the induction of Whittaker models is proved for Bessel models which are minimal in a suitable sense. The holomorphicity in the induction parameter of the Bessel functional is established. Last, local coefficients are defined for each irreducible supercuspidal representation which carries a Bessel functional and also for a certain component of each representation parabolically induced from such a supercuspidal.
We examine the theory of induced representations for non-connected reductive $p$-adic groups for which $G/G^0$ is abelian. We first examine the structure of those representations of the form $\Ind_{P^0}^G(σ),$ where $P^0$ is a parabolic subgroup of $G^0$ and $\s$ is a discrete series representation of the Levi component of $P^0.$ Here we develop a theory of $R$--groups, extending the theory in the connected case. We then prove some general results in the theory of representations of non-connected $p$-adic groups whose component group is abelian. We define the notion of cuspidal parabolic for $G$ in order to give a context for this discussion. Intertwining operators for the non-connected case are examined and the notions of supercuspidal and discrete series are defined. Finally, we examine parabolic induction from a cuspidal parabolic subgroup of $G.$ Here we also develop a theory of $R$--groups, and show that these groups parameterize the induced representations in a manner that is consistent with the connected case and with the first set of results as well.