On diagonal dominance arguments for bounding 6A-16∞

In a recent paper by J.M. Varah, an upper bound for 6A^-16_∞ was determined, under the assumption that A is strictly diagonally dominant, and this bound was then used to obtain a lower bound for the smallest singular value for A. In this note, this upper bound for 6A^-16_∞ is sharpened, and extended to a wider class of matrices. This bound is then used to obtain an improved lower bound for the smallest singular value of a matrix.     

Reliability and expectation bounds for coherent systems with exchangeable components

Sharp upper and lower bounds are obtained for the reliability functions and the expectations of lifetimes of coherent systems based on dependent exchangeable absolutely continuous components with a given marginal distribution function, by use of the concept of Samaniego's signature. We first show that the distribution of any coherent system based on exchangeable components with absolutely continuous joint distribution is a convex combination of distributions of order statistics (equivalent to the k-out-of-n systems) with the weights identical with the values of the Samaniego signature of the system. This extends the Samaniego representation valid for the case of independent and identically distributed components. Combining the representation with optimal bounds on linear combinations of distribution functions of order statistics from dependent identically distributed samples, we derive the corresponding reliability and expectation bounds, dependent on the signature of the system and marginal distribution of dependent components. We also present the sequences of exchangeable absolutely continuous joint distributions of components which attain the bounds in limit. As an application, we obtain the reliability bounds for all the coherent systems with three and four exchangeable components, expressed in terms of the parent marginal reliability function and specify the respective expectation bounds for exchangeable exponential components, comparing them with the lifetime expectations of systems with independent and identically distributed exponential components.     

Estimates for the finite-time ruin probability with insurance and financial risks

The paper gives estimates for the finite-time ruin probability with insurance and financial risks. When the distribution of the insurance risk belongs to the class L(??) for some ?? > 0 or the subexponential distribution class, we abtain some asymptotic equivalent relationships for the finite-time ruin probability, respectively. When the distribution of the insurance risk belongs to the dominated varying-tailed distribution class, we obtain asymptotic upper bound and lower bound for the finite-time ruin probability, where for the asymptotic upper bound, we completely get rid of the restriction of mutual independence on insurance risks, and for the lower bound, we only need the insurance risks to have a weak positive association structure. The obtained results extend and improve some existing results.     

Packing the maximum number of m × n tiles in a large p × q rectangle

For relatively prime m and n we determine precisely the wasted area when a large p × q rectangle is packed with m × n tiles in the most efficient manner. The case m = 1 is considered first, and we derive a formula for the wasted area depending only on the residue classes of p and q (mod n). This result has also been obtained by various other authors. Then by regarding an m × n tile as a union of 1 × n or 1 × m tiles a lower bound for the wasted area is obtained. Finally, by a series of explicit constructions, we show that for sufficiently large p, q, this lower bound is actually the correct value.     

Finite Exchangeability, Lévy-Frailty Copulas and Higher-Order Monotonic Sequences

This paper deals with a surprising connection between exchangeable distributions on {0,1} ⁿ and the recently introduced Lévy-frailty copulas, the link being provided by a new class of multivariate distribution functions called linearly order symmetric. The characterisation theorem for Lévy-frailty copulas is given a new and short (non-combinatorial) proof, and a related result is shown for exchangeable Marshall–Olkin distributions. A common thread in all these considerations is higher-order monotonic functions on integer intervals of the form {0,1,…,n}.     

Some results for the queues Mx/M/c and GIx/ G/c

We develop for the queue M^x/M/c an upper bound for the mean queue length and lower bounds for the delay probabilities (that of an arrival group and that of an arbitrary customer in the arrival group). An approximate formula is also developed for the general bulk-arrival queue GI^x/G/c. Preliminary numerical studies have indicated excellent performance of the results.     

Assessing process capability based on the lower confidence bound of Cpk for asymmetric tolerances

It has been proved that process capability indices provide very efficient measures of the capability of processes from many different perspectives. At the present time, the C_pk index is used more than any other index for measuring process capability. However, most existing research works for capability testing have focused on processes with symmetric tolerances, but not for asymmetric tolerances. A lower confidence bound estimates the minimum process capability, conveying critical information regarding product quality, which is essential to quality assurance. The sample size determination, which provides the sample sizes necessary to achieve a desired lower confidence bound, is directly related to the cost of the data collection plan. This paper provides explicit formulas with efficient algorithms to obtain the lower confidence bounds and sample sizes required for specified precision of the estimation on C_pk for processes with asymmetric tolerances. A Matlab computer program using a binary search method is developed. For the practitioners to use in their in-plant applications, we tabulate lower confidence bounds for some commonly used capability requirement and the sampling accuracy of C_pk for sample sizes determination. A realistic example of forging process is presented to illustrate the applicability of the proposed method.     

On the number of solutions of N − p = P3

A lower bound of Richert on the number of solutions of N ? p = P₃ is improved.     

Joint distribution of run statistics in partially exchangeable processes

Let {X_i}_i≥1 be an infinite sequence of recurrent partially exchangeable random variables with two possible outcomes as either “1” (success) or “0” (failure). In this paper we obtain the joint distribution of success and failure run statistics in {X_i}_i≥1. The results can be used to obtain the joint distribution of runs in ordinary Markov chains, exchangeable and independent sequences.     

Perfect codes in the graphs Ok

In this paper we consider the existence of perfect codes in the infinite class of distance-transitive graphs O_k. Perfect 1-codes correspond to certain Steiner systems and necessary conditions for the existence of such a code are satisfied if k + 1 is prime. We give some nonexistence results for perfect 2-, 3-, and 4-codes and for perfect e-codes in general, including a lower bound for k in terms of e.     

Distribution of Artin–Schreier extensions

The article at hand contains exact asymptotic formulas for the distribution of conductors of elementary abelian p-extensions of global function fields of characteristic p. As a consequence for the distribution of discriminants, this leads to an exact asymptotic formula for simple cyclic extensions and an interesting lower bound for noncyclic elementary abelian extensions.     

Indépendance Linéaire sur Q de logarithmes P-adiques de nombres algébriques et rang P-adique du groupe des unités d'un corps de nombres

Following Ax's method a lower bound for the p-adic rank of the group of units in the general case of a Galois number field over the rationals is given. In some nontrivial special cases the result gives Leopoldt's conjecture. The same method is also applied to the case of p-units.     

Growth and poles of meromorphic solutions of some difference equations

This paper is devoted to studying the growth property and the pole distribution of meromorphic solutions f of some complex difference equations with all coefficients being rational functions or of growth S(r,f). We find the lower bound of the lower order, or the relation between lower order and the convergence exponent of poles of meromorphic solutions of such equations.     

Smooth and discontinuous junctions in the p-system

Consider the p-system describing the subsonic flow of a fluid in a pipe with section a=a(x). We prove that the resulting Cauchy problem generates a Lipschitz semigroup, provided the total variation of the initial datum and the oscillation of a are small. An explicit estimate on the bound of the total variation of a is provided, showing that at lower fluid speeds, higher total variations of a are acceptable. An example shows that the bound on TV(a) is mandatory, for otherwise the total variation of the solution may grow arbitrarily.     

On p-adic zeros of forms

Using the procedure of G. I. Arkhipov and A. A. Karatsuba (Math. USSR-Izv.19 (1982), 321–340), their exponential lower bound on the number of variables possible for a form of degree d having only the trivial p-adic zero is sharpened.     

Bounds on the number of numerical semigroups of a given genus

Lower and upper bounds are given for the number n_g of numerical semigroups of genus g. The lower bound is the first known lower bound while the upper bound significantly improves the only known bound given by the Catalan numbers. In a previous work the sequence n_g is conjectured to behave asymptotically as the Fibonacci numbers. The lower bound proved in this work is related to the Fibonacci numbers and so the result seems to be in the direction to prove the conjecture. The method used is based on an accurate analysis of the tree of numerical semigroups and of the number of descendants of the descendants of each node depending on the number of descendants of the node itself.     

Bounds on the Rubbling and Optimal Rubbling Numbers of Graphs

A pebbling move on a graph removes two pebbles at a vertex and adds one pebble at an adjacent vertex. Rubbling is a version of pebbling where an additional move is allowed. In this new move, one pebble each is removed at vertices v and w adjacent to a vertex u, and an extra pebble is added at vertex u. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using rubbling moves. The rubbling number is the smallest number m needed to guarantee that any vertex is reachable from any pebble distribution of m pebbles. The optimal rubbling number is the smallest number m needed to guarantee a pebble distribution of m pebbles from which any vertex is reachable. We give bounds for rubbling and optimal rubbling numbers. In particular, we find an upper bound for the rubbling number of n-vertex, diameter d graphs, and estimates for the maximum rubbling number of diameter 2 graphs. We also give a sharp upper bound for the optimal rubbling number, and sharp upper and lower bounds in terms of the diameter.     

On k-planar crossing numbers

The k-planar crossing number of a graph is the minimum number of crossings of its edges over all possible drawings of the graph in k planes. We propose algorithms and methods for k-planar drawings of general graphs together with lower bound techniques. We give exact results for the k-planar crossing number of K_2k+1,q, for k?2. We prove tight bounds for complete graphs. We also study the rectilinear k-planar crossing number.     

On Stein?s method for infinite-dimensional Gaussian approximation in abstract Wiener spaces

In this paper, we generalize Stein?s method to “infinite-variate” normal approximation that is an infinite-dimensional approximation by abstract Wiener measures on a real separable Banach space. We first establish a Stein?s identity for abstract Wiener measures and solve the corresponding Stein?s equation. Then we will present a Gaussian approximation theorem using exchangeable pairs in an infinite-variate context. As an application, we will derive an explicit error bound of Gaussian approximation to the distribution of a sum of independent and identically distributed Banach space-valued random variables based on a Lindeberg-Lévy type limit theorem. In addition, an analogous of Berry-Esséen type estimate for abstract Wiener measures will be obtained.     

Inequalities for characteristic functions of multidimensional distributions

We obtain lower and upper bounds for the absolute values of characteristic functions of multivariate distributions F and also derive a lower bound on the norm of the zeroes of a characteristic function in terms of moments of the norm of the random vector with distribution F. Similar results are obtained for characteristic functions of probability measures on a separable Hilbert space.