An exact test of the accuracy of binary classification models based on the probability distribution of the average rank

We propose a new way to evaluate the discriminatory power of models that generate a continuous value as the basis for performing a binary classification task. Our hypothesis test uses the average rank of the k successes in the sample of size n, based on those continuous values. We derive the probability mass function for the average rank from the coefficients of a Gaussian polynomial distribution that results from randomly sampling k distinct positive integers, all n or less. The significance level of the test is found by counting the number of arrangements that produce average ranks more extreme than the one observed. Recursive relationships can be used to calculate the values necessary to compute the p-value. For large values of k and n, for which exact computation might be prohibitive, we present numerical results which indicate that the critical values of the distribution are nearly linear in n for a fixed k and that the coefficients of the linear relationships are nonlinear functions of k and the desired percentile. We develop regression models for those relationships to approximate the number of arrangements in order to make the test practical for large values of k and n.     

On the boundedness of the total variation of the logarithm of a Blaschke product

We establish that, for a Blaschke product B(z) convergent in the unit disk, the condition - ∞ < \smallint ₀¹ log(1 - t)n(t,B)dt\smallint _0^1 \log (1 - t)n(t,B)dt is sufficient for the total variation of logB to be bounded on a circle of radiusr, 0 <r < 1. For products B(z) with zeros concentrated on a single ray, this condition is also necessary. Here, n(t, B) denotes the number of zeros of the functionB (z) in a disk of radiust.     

Distribution of large values of the argument of the Riemann zeta function on short intervals

An upper bound for the measure of the set of values t ∈ (T,T + H] for H = T ^27/82+ɛ for which |S(t)| ≥ λ is obtained.     

On the number of list-colorings

Given a list of boxes L for a graph G (each vertex is assigned a finite set of colors that we call a box), we denote by f(G, L) the number of L-colorings of G (each vertex must be colored wiht a color of its box). In the case where all the boxes are identical and of size k, f(G, L) = p(G, k), where P=G, k) is the chromatic polynominal of G. We denote by F(G, k) the minimum of f(G, L) over all the lists of boxes such that each box has size at least k. It is clear that F(G, k) ≤ P(G, k) for all G, k, and we will see in the introduction some examples of graphs such that F(G, k) < P(G, k) for some k. However, we will show, in answer to a problem proposed by A. Kostochka and A. Sidorenko (Fourth Czechoslovak Symposium on Combinatorics, Prachatice, Jin, 1990), that for all G, F(G, k) = P(G, k) for all k sufficiently large. It will follow in particular that F(G, k) is not given by a polynominal in k for all G. The proof is based on the analysis of an algorithm for computing f(G, L) analogous to the classical one for computing P(G, k).     

Weak Asymptotics for the Generating Polynomials of the Stirling Numbers of the Second Kind

For the horizontal generating functions P_n(z)=∑ⁿ_k=1 S(n, k) z^k of the Stirling numbers of the second kind, strong asymptotics are established, as n→∞. By using the saddle point method for Q_n(z)=P_n(nz) there are two main results: an oscillating asymptotic for z(−e, 0) and a uniform asymptotic on every compact subset of \[−e, 0]. Finally, an Airy asymptotic in the neighborhood of −e is deduced.     

Estimates of the conformal scalar curvature equation via the method of moving planes

In this paper we derive a local estimate of a positive singular solution u near its singular set Z of the conformal equation where K(x) is a positive continuous function, Z is a compact subset of , and g satisfies that is nonincreasing for t > 0. Assuming that the order of flatness at critical points of K on Z is no less than , we prove that, through the application of the method of moving planes, the inequality holds for any solution of (0.1) with Cap(Z) = 0. By the same method, we also derive a Harnack-type inequality for smooth positive solutions. Let u satisfy Assume that the order of flatness at critical points of K is no less than n - 2; then the inequality holds for R ≤ 1. We also show by examples that the assumption about the flatness at critical points is optimal for validity of the inequality (0.4). © 1997 John Wiley & Sons, Inc.     

The behavior of the Laplace transform of the invariant measure on the hypersphere of high dimension

We consider the sequence of the hyperspheres M _n , i.e., the homogeneous transitive spaces of the Cartan subgroup of the group and study the normalized limit of the corresponding sequence of invariant measures m _n on those spaces. In the case of compact groups and homogeneous spaces, for example, for the classical pairs (SO(n), S ^n-1), n = 1, 2, … , the limit of the corresponding measures is the classical infinite-dimensional Gaussian measure; this is the well-known Maxwell-Poincaré lemma. Simultaneously the Gaussian measure is a unique (up to a scalar) invariant measure with respect to the action of the infinite orthogonal group O(∞). This coincidence implies the asymptotic equivalence between grand and small canonical ensembles for the series of the pairs (SO(n), S ^n-1). Our main result shows that the situation for noncompact groups, for example for the case , is completely different: the limit of the measures m _n does not exist in the literal sense, and we show that only a normalized logarithmic limit of the Laplace transforms of those measures does exist. At the same time, there exists a measure which is invariant with respect to a continuous analogue of the Cartan subgroup of the group GL(∞), the so-called infinite-dimensional Lebesgue measure (see [7]). This difference is an evidence for non-equivalence between the grand and small canonical ensembles in the noncompact case. To my friend Dima Arnold     

On the number of nonzero digits of the partition function

Let p(n) be the function that counts the number of partitions of n. Let b ≥ 2 be a fixed positive integer. In this paper, we show that for almost all n the sum of the digits of p(n) in base b is at least log n/(7log log n). Our proof uses the first term of Rademacher’s formula for p(n).     

On the hopficity of the polynomial rings

In this paper we prove that if a ringR satisfies the condition that for some integern > 1,a ⁿ =a for everya inR, thenR a hopfian ring implies that the ringR [T] of polynomials is also hopfian. This generalizes a recent result of Varadarajan which states that ifR is a Boolean hopfian ring then the ringR[T] is also hopfian. We show furthermore that there are numerous ringsR satisfying the hypothesis of our theorem which are neither Boolean nor Noetherian.     

Interpolation properties in the extensions of the logic of inequality

We consider the modal logics wK4 and DL as well as the corresponding weakly transitive modal algebras and DL-algebras. We prove that there exist precisely 16 amalgamable varieties of DL-algebras. We find a criterion for the weak amalgamation property of varieties of weakly transitive modal algebras, solve the deductive interpolation problem for extensions of the logic of inequality DL, and obtain a weak interpolation criterion over wK4.     

On the dimension of the module-varieties of tame and weld algebras

Inspired by a result in [Ga], we locate three integer forms of F_q[SL(n+ 1)] over k[q,q ^-1] wih a presentation by generators and relations, which for q=1 specialize to U(𝔥)), where 𝔥 is the Lie bialgebra of the Poisson Lie group dual to SL(n+1). In sight of this we prove two PBW-like theorems for F_q [SL(n+ 1)], both related to the classical PBW theorem for U(𝔥).     

Symmetry properties for the extremals of the Sobolev trace embedding

In this article we study symmetry properties of the extremals for the Sobolev trace embedding H¹(B(0,μ))L^q(∂B(0,μ)) with 1q2(N−1)/(N−2) for different values of μ. These extremals u are solutions of the problem

We find that, for 1q<2(N−1)/(N−2), there exists a unique normalized extremal u, which is positive and has to be radial, for μ small enough. For the critical case, q=2(N−1)/(N−2), as a consequence of the symmetry properties for small balls, we conclude the existence of radial extremals. Finally, for 1<q2, we show that a radial extremal exists for every ball.     

A New Lower Bound on the Number of Odd Values of the Ordinary Partition Function

The parity of p(n), the ordinary partition function, has been studied for at least a century, yet it still remains something of a mystery. Although much work has been done, the known lower bounds for the number of even and odd values of p(n) for n ≤ N still appear to have a great deal of room for improvement. In this paper, we use classical methods to give a new lower bound for the number of odd values of p(n).     

On the upper bounds of the minimum number of rows of disjunct matrices

A 0-1 matrix is d-disjunct if no column is covered by the union of any d other columns. A 0-1 matrix is (d; z)-disjunct if for any column C and any d other columns, there exist at least z rows such that each of them has value 1 at column C and value 0 at all the other d columns. Let t(d, n) and t(d, n; z) denote the minimum number of rows required by a d-disjunct matrix and a (d; z)-disjunct matrix with n columns, respectively. We give a very short proof for the currently best upper bound on t(d, n). We also generalize our method to obtain a new upper bound on t(d, n; z). The work of Y. Cheng and G. Lin is supported by Natural Science and Engineering Research Council (NSERC) of Canada, and the Alberta Ingenuity Center for Machine Learning (AICML) at the University of Alberta. The work of D.-Z. Du is partially supported by National Science Foundation under grant No.CCF0621829.     

Morita Invariance of the Filter Dimension and of the Inequality of Bernstein

It is proved that the filter dimension is Morita invariant. A direct consequence of this fact is the Morita invariance of the inequality of Bernstein: if an algebra A is Morita equivalent to the ring of differential operators on a smooth irreducible affine algebraic variety X of dimension n ≥ 1 over a field of characteristic zero then the Gelfand–Kirillov dimension for all nonzero finitely generated A-modules M. In fact, a stronger result is proved, namely, a Morita invariance of the holonomic number for finitely generated algebra. A direct consequence of this fact is that an analogue of the inequality of Bernstein holds for the (simple) rational Cherednik algebras H _c for integral c: for all nonzero finitely generated H _c -modules M. For these class of algebras, it gives an affirmative answer to a question of Ken Brown about symplectic reflection algebras. Presented by Alain Verschoren.     

On the difference property of the family of functions with the Baire property

We show that it is consistent with ZFC that the family of functions with the Baire property has the difference property. That is, every function for which f(x + h)-f(x) has the Baire property for every h∈R is of the form f=g + Awhere g has the Baire property and A is additive. This revised version was published online in June 2006 with corrections to the Cover Date.     

Sections of the Difference Body

Let K be an n -dimensional convex body. Define the difference body by K-K= { x-y | x,y ∈ K }. We estimate the volume of the section of K-K by a linear subspace F via the maximal volume of sections of K parallel to F . We prove that for any m -dimensional subspace F there exists x ∈ \bf R ⁿ , such that for some absolute constant C . We show that for small dimensions of F this estimate is exact up to a multiplicative constant. Received May 6, 1998, and in revised form July 23, 1998.     

On the approximability of the maximum induced matching problem

In this paper we consider the approximability of the maximum induced matching problem (MIM). We give an approximation algorithm with asymptotic performance ratio d−1 for MIM in d-regular graphs, for each d3. We also prove that MIM is APX-complete in d-regular graphs, for each d3.     

On the sensitivity of the LU factorization

This paper gives sensitivity analyses by two approaches forL andU in the factorizationA=LU for general perturbations inA which are sufficiently small in norm. By the matrix-vector equation approach, we derive the condition numbers for theL andU factors. By the matrix equation approach we derive corresponding condition estimates. We show how partial pivoting and complete pivoting affect the sensitivity of the LU factorization. The material presented here is a part of the first author's PhD thesis under the supervision of the second author. This research was supported by NSERC of Canada Grant OGP0009236.