a regression model for multivariate random length data multivariate random length data occur when we observe multiple measurements of a quantitative variable and the variable number of these measurements is also an observed outcome for each experimental unit for example for a patient with coronary artery disease we may observe a number of lesions in that patient s coronary arteries along with percentage of blockage of each lesion barnhart and sampson first proposed the multiple population model to analyse multivariate random length data without covariates this paper extends their approach to deal with multiple covariates we propose a new multiple population regression model with covariates and discuss the estimation issues we analyse data from the type ii coronary intervention study to illustrate the methodology

on a decomposition lemma for positive semi definite block matrices abstract this short note in part of expository nature points out several new or recent consequences of a quite nice decomposition for positive semi definite matrices

on the efficiency of pairing heaps and related data structures the pairing heap is well regarded as an efficient data structure for implementing priority queue operations it is included in the gnu c library strikingly simple in design the pairing heap data structure nonetheless seems difficult to analyze belonging to the genre of self adjusting data structures with its design originating as a self adjusting analogue of the fibonacci heap it has been previously conjectured that the pairing heap provides constrant amortized time decrease key operations and experimental studies have supported this conjecture this paper demonstrates contrary to conjecture that the pairing heap requires more than constant amortized time to perform decrease key operations moreover new experimental findings are presented that reveal detectable growth in the amortized cost of the decrease key operation second a unifying framework is developed that includes both pairing heaps and fibonacci heaps the parameter of interest in this framework is the storage capacity available in the nodes of the data structure for auxiliary balance information fields in this respect fibonacci heaps require log log n bits per node when n items are present this is shown to be asymptotically optimal for data structures that achieve the same asymptotic performance bounds as fibonacci heaps and fall within this framework

logarithmic lower bounds in the cell probe model we develop a new technique for proving cell probe lower bounds on dynamic data structures this technique enables us to prove an amortized randomized omega lg n lower bound per operation for several data structural problems on n elements including partial sums dynamic connectivity among disjoint paths or a forest or a graph and several other dynamic graph problems by simple reductions such a lower bound breaks a long standing barrier of omega lg n lg lg n for any dynamic language membership problem it also establishes the optimality of several existing data structures such as sleator and tarjan s dynamic trees we also prove the first omega log b n lower bound in the external memory model without assumptions on the data structure such as the comparison model our lower bounds also give a query update trade off curve matched e g by several data structures for dynamic connectivity in graphs we also prove matching upper and lower bounds for partial sums when parameterized by the word size and the maximum additive change in an update

a joint modeling approach for multivariate survival data with random length summary in many biomedical studies that involve correlated data an outcome is often repeatedly measured for each individual subject along with the number of these measurements which is also treated as an observed outcome this type of data has been referred as multivariate random length data by barnhart and sampson 1995 a common approach to handling such type of data is to jointly model the multiple measurements and the random length in previous literature a key assumption is the multivariate normality for the multiple measurements motivated by a reproductive study we propose a new copula based joint model which relaxes the normality assumption specifically we adopt the clayton oakes model for multiple measurements with flexible marginal distributions specified as semi parametric transformation models the random length is modeled via a generalized linear model we develop an approximate em algorithm to derive parameter estimators and standard errors of the estimators are obtained through bootstrapping procedures and the finite sample performance of the proposed method is investigated using simulation studies we apply our method to the mount sinai study of women office workers msswow where women were prospectively followed for 1 year for studying fertility

on the dynamic finger conjecture for splay trees part ii the proof the following result is shown on an n node splay tree the amortized cost of an access at distance d from the preceding access is o log d 1 in addition there is an o n initialization cost the accesses include searches insertions and deletions

better analysis of binary search tree on decomposable sequences abstract in their seminal paper sleator and tarjan 1985 15 the authors conjectured that the splay tree is a dynamically optimal binary search tree bst in spite of decades of intensive research the problem remains open perhaps a more basic question which has also attracted much attention is if there exists any dynamically optimal bst algorithm one such candidate is which is a simple and intuitive bst algorithm lucas 1988 11 munro 2000 12 demaine et al 2009 6 demaine et al 2009 6 showed a novel connection between a geometric problem and the binary search tree problem related to the above conjecture however there has been little progress in solving this geometric problem too since dynamic optimality conjecture in its most general form remains elusive despite much effort researchers have studied this problem on special sequences recently chalermsook et al 2015 2 studied a type of sequences known as k decomposable sequences in this context where k parametrizes easiness of the sequence using tools from forbidden submatrix theory they showed that takes n 2 o k 2 time on this sequence and explicitly raised the question of improving this bound in this paper we show that takes o n log k time on k decomposable sequences in contrast to the previous approach ours is based on first principles one of the main ingredients of our result is a new construction of a lower bound certificate on the performance of any algorithm this certificate is constructed using the execution of and is more nuanced and possibly more flexible than the previous independent set certificate of demaine et al this result which is applicable to all sequences may be of independent interest and may lead to further progress in analyzing on k decomposable as well as general sequences

optimal biweighted binary trees and the complexity of maintaining partial sums let a be an array the partial sum problem concerns the design of a data structure for implementing the following operations the operation update j x has the effect a j leftarrow a j x and the query operation ssum j returns the partial sum sum i 1 j a i our interest centers upon the optimal efficiency with which sequences of such operations can be performed and we derive new upper and lower bounds in the semigroup model of computation our analysis relates the optimal complexity of the partial sum problem to optimal binary trees relative to a type of weighting scheme that defines the notion of biweighted binary tree

dynamic optimality almost we present an o lg lg n competitive online binary search tree improving upon the best previous trivial competitive ratio of o lg n this is the first major progress on sleator and tarjan s dynamic optimality conjecture of 1985 that o 1 competitive binary search trees exist

lower bounds for accessing binary search trees with rotations two methods are given for obtaining lower bounds on the cost of accessing a sequence of nodes in a symmetrically ordered binary search tree in a model where rotations can be done on the tree and the entire sequence is known before accessing begins but the accesses must be done in the order given for example it can be proven that the bit reversal permutation requires theta n log n time to access in this model it is also shown that the expected cost of accessing random sequences in this model is the same as it is for the case where the tree is static

the nonexistence of 100 1 alpha confidence sets of finite expected diameter in errors in variables and related models on montre que pour certains modeles il est impossible de construire des intervalles de confiance pour les parametres cles qui aient a la fois une confiance positive et une longueur esperee finie

a unified access bound on comparison based dynamic dictionaries we present a dynamic comparison based search structure that supports insertions deletions and searches within the unified bound the unified bound specifies that it is quick to access an element that is near a recently accessed element more precisely if w y distinct elements have been accessed since the last access to element y and d x y denotes the rank distance between x and y among the current set of elements then the amortized cost to access element x is o minylog w y d x y 2 this property generalizes the working set and dynamic finger properties of splay trees

determinantal and eigenvalue inequalities for matrices with numerical ranges in a sector abstract let a a 11 a 12 a 21 a 22 m n where a 11 m m with m n 2 be such that the numerical range of a lies in the set e i z c z z tan for some 0 2 and 0 2 we obtain the optimal containment region for the generalized eigenvalue satisfying a 11 0 0 a 22 x 0 a 12 a 21 0 x for some nonzero x c n and the optimal eigenvalue containment region of the matrix i m a 11 1 a 12 a 22 1 a 21 in case a 11 and a 22 are invertible from this result one can show det a sec 2 m det a 11 det a 22 in particular if a is an accretive dissipative matrix then det a 2 m det a 11 det a 22 these affirm some conjectures of drury and lin

a study on splay trees abstract we study the dynamic optimality conjecture which predicts that splay trees are a form of universally efficient binary search tree for any access sequence we reduce this claim to a regular access bound which seems plausible and might be easier to prove this approach may be useful to establish dynamic optimality

self adjusting binary search trees the splay tree a self adjusting form of binary search tree is developed and analyzed the binary search tree is a data structure for representing tables and lists so that accessing inserting and deleting items is easy on an n node splay tree all the standard search tree operations have an amortized time bound of o log n per operation where by amortized time is meant the time per operation averaged over a worst case sequence of operations thus splay trees are as efficient as balanced trees when total running time is the measure of interest in addition for sufficiently long access sequences splay trees are as efficient to within a constant factor as static optimum search trees the efficiency of splay trees comes not from an explicit structural constraint as with balanced trees but from applying a simple restructuring heuristic called splaying whenever the tree is accessed extensions of splaying give simplified forms of two other data structures lexicographic or multidimensional search trees and link cut trees

fischer type determinantal inequalities for accretive dissipative matrices abstract let a a 11 a 12 a 21 a 22 be an n n accretive dissipative matrix k and l be the orders of a 11 and a 22 respectively and let m min k l then det a a det a 11 det a 22 where a 2 3 m 2 if m n 3 2 n 2 if n 3 m n 2 this improves a result of ikramov

the cost of offline binary search tree algorithms and the complexity of the request sequence in evaluating the performance of online algorithms for search trees one wants to compare them to the best offline algorithm available in this paper we lower bound the cost of an optimal offline binary search tree using the kolmogorov complexity of the request sequence we obtain several applications for this result first any offline binary search tree algorithm can be at most a constant factor away from the entropy of the process producing the request sequence second for a fraction 1 1 2 m of request sequences of length m on n items the cost of any offline algorithm is w m logn 1 third the expected cost of splay trees is within a constant factor of the expected cost of an optimal offline binary search tree algorithm in a subset of markov chains

multiple population models for multivariate random length data with applications in clinical trials this paper focuses on the development and study of multiple population models for multivariate random length data of the type often encountered in clinical trials if experimental outcomes per subject consist of multiple measurements of a quantitative variable and the number of these measurements then a multivariate random length vector is observed for this type of data the experimental treatment is likely to affect both the quantitative measurements and the number of these measurements one example of such data is from the national heart lung and blood institute type ii coronary intervention study brensike et al 1982 controlled clinical trials 3 91 111 1984 circulation 69 313 324 the outcome data consist of vectors of lesion sizes with lengths determined by the number of underlying lesions assessed from the patients angiograms where both the numbers and the lesion sizes depend on patients overall disease status we propose models which can realistically describe the relationships between the quantitative variables and the number of responses the asymptotic covariance of the maximum likelihood estimators is obtained data from the type ii study are analyzed using this multiple population model

norm inequalities for accretive dissipative operator matrices abstract an operator on a hilbert space is accretive dissipative if its real and imaginary parts are both positive in this paper we consider norm inequalities for an accretive dissipative operator matrix the inequalities obtained extend some known results

norm inequalities for sector block matrices abstract we study unitarily invariant norm inequalities for sector block matrices and obtain new inequalities which extend some recent results of jabbarzadeh and kaleibary 2020 and generalize a norm inequality due to lin and zhou 2013 at the same time we refine an inequality raised by zhang 2015

weak majorization inequalities and convex functions abstract let f be a convex function defined on an interval i 0 1 and a b n n complex hermitian matrices with spectrum in i we prove that the eigenvalues of f a 1 b are weakly majorized by the eigenvalues of f a 1 f b further if f is log convex we prove that the eigenvalues of f a 1 b are weakly majorized by the eigenvalues of f a f b 1 as applications we obtain generalizations of the famous golden thomson trace inequality a representation theorem and a harmonic geometric mean inequality some related inequalities are discussed

norm inequalities involving accretive dissipative 2 2 block matrices abstract let t 11 t 12 t 21 and t 22 be n n complex matrices and let t t 11 t 12 t 21 t 22 be accretive dissipative it is shown that if f is an increasing convex function on 0 such that f 0 0 then f t 12 2 f t 21 2 f t 2 for every unitarily invariant norm moreover if f is an increasing concave function on 0 such that f 0 0 then f t 12 2 f t 21 2 4 f t 2 4 for every unitarily invariant norm among other inequalities for the schatten p norms it is shown that t 12 p p t 21 p p 2 p 1 t 11 p p 2 t 22 p p 2 for p 2