Roughly Speaking Research Articles

Overdetermined elliptic problems in onduloid-type domains with general nonlinearities

In this paper, we prove the existence of nontrivial unbounded domains Ω⊂Rn+1,n≥1, bifurcating from the straight cylinder B×R (where B is the unit ball of Rn), such that the overdetermined elliptic problem{Δu+f(u)=0in Ω, u=0on ∂Ω, ∂νu=constanton ∂Ω, has a positive bounded solution. We will prove such result for a very general class of functions f:[0,+∞)→R. Roughly speaking, we only ask that the Dirichlet problem in B admits a nondegenerate solution. The proof uses a local bifurcation argument.

“Roughly Speaking”: Why Do U.S. Foresters Measure DBH at 4.5 Feet?

This paper traces the history of diameter at breast height (DBH) tree measurements in the United States and suggests that DBH is a socially constructed scientific standard. In 1899, Gifford Pinchot—arguably the most important forester in U.S. history—claimed that 4.5 feet (ft) (137 cm) was the customary height for measuring the diameter of a tree’s trunk, which then became the default height in the United States. However, based on my review of archival forestry texts, there was wide variation in the heights used to measure trees pre-1900. Furthermore, my analysis of male anthropometric datasets contradicts assertions that 4.5 ft is a comfortable measurement height for men. Rather, 4.5 ft was likely Pinchot’s own BH. The selection of 4.5 ft might have also resulted from a mis-conversion of Bavarian forestry data. I discuss the social context in which the 4.5 ft standard emerged.

고전의 역사성과 고전번역의 어려움

There seem to be three defining characteristics of being a classic. (1) Universality: Classics deal with universal questions; their value transcends individual societies. (2) Fundamental reflection: Classics do not turn away from the problems of the given society; they rather go deeper to the roots of the problems. (3) Forming the framework of thinking: Classics help people to form and reshape their framework of thinking. And these defining characteristics of classics are sources of special difficulties in translating them.BR Roughly speaking, different languages presuppose different world-views, and this is what makes translation in general a difficult task. But the three characteristics above amplify the difficulties in translating classics. Differences in world-views are more clearly revealed when we are dealing with universal questions. Similarly, fundamental reflections make the differences in world-views more visible. Moreover, fundamental reflections tend to touch on the core of a given world-view, and may cause slight changes in it. Cumulative changes in the world-view result in the change of the framework of thoughts itself. And the changes in the framework of thoughts itself may even cause a change in the grammatical structure of the language. To sum up, classics have historicality, and that makes translation of classics a particularly difficult task.BR I discuss two cases of the difficulties in translating classics: sōphrosunē and third-person imperative. Sōphrosunē was a very important concept in the Greek religion, as embodied in the delphic maxim, “Know thyself”. Plato seems to have reflected fundamentally and continually on this concept. In the Charmides, he takes it as knowledge of what one knows and what one does not know. In the Protagoras, he takes it, in effect, as knowledge of goodness and badness. And in the Republic, he takes it as the agreement between the parts of the soul concerning which part should rule and which part should be ruled. And there is simply no word that can correctly translate sōphrosunē that covers all these conceptions.BR As to the third-person imperative, while there were abundant usages of it in Ancient Greek, the very idea of giving orders to a third-person seems illogical to most of us modern people. But it seems to me that there are third-person imperatives in Korean as well. Although grammarians deny its existence and try to give alternative explanations for sentences that look like third-person imperatives, I argue that there are such sentences that cannot be explained away in that way. The upshot is that you should not translate Greek third-person imperatives automatically as ‘Let X do Y’.

Complexity dichotomies for the Minimum[formula omitted]-Overlay problem

For a (possibly infinite) fixed family of graphs F, we say that a graph G overlaysF on a hypergraph H if V(H) is equal to V(G) and the subgraph of G induced by every hyperedge of H contains some member of F as a spanning subgraph. While it is easy to see that the complete graph on |V(H)| overlays F on a hypergraph H whenever the problem admits a solution, the MinimumF-Overlay problem asks for such a graph with at most k edges, for some given k∈N. This problem allows to generalize some natural problems which may arise in practice. For instance, if the family F contains all connected graphs, then MinimumF-Overlay corresponds to the Minimum Connectivity Inference problem (also known as Subset Interconnection Design problem) introduced for the low-resolution reconstruction of macro-molecular assembly in structural biology, or for the design of networks.Our main contribution is a strong dichotomy result regarding the polynomial vs. NP-complete status with respect to the considered family F. Roughly speaking, we show that the easy cases one can think of (e.g. when edgeless graphs of the right sizes are in F, or if F contains only cliques) are the only families giving rise to a polynomial problem: all others are NP-complete. We then investigate the parameterized complexity of the problem and give similar sufficient conditions on F that give rise to W[1]-hard, W[2]-hard or FPT problems when the parameter is the size of the solution. This yields an FPT/W[1]-hard dichotomy for a relaxed problem, where every hyperedge of H must contain some member of F as a (non necessarily spanning) subgraph.

Computing the coarseness with strips or boxes

Recently, the concept of coarseness was introduced as a measure of how blended a 2-colored point set S is. In the definition of this measure, a convex partition Π, that is, a partition of S into sets {S1,…,Sk} of S whose convex hulls are pairwise disjoint, is considered. The discrepancy of Π, denoted by d(S,Π), is the smallest (bichromatic) discrepancy of the elements of Π. The coarseness of S, denoted by C(S), is then defined as the maximum of d(S,Π) over all convex partitions Π of S. Roughly speaking, the value of the coarseness is high when we can split S into blocks, each with large discrepancy. It has been conjectured that computing the coarseness is NP-hard. In this paper, we study how to compute the coarseness for two constrained cases: (1) when the k elements of Π are separated by k−1 pairwise parallel lines (strips) and, (2) the case in which the cardinality of the partition is fixed and the elements of Π are covered by pairwise disjoint axis-aligned rectangles (boxes). For the first case we present an O(n2log2n)-time algorithm, and show that such a computation problem is 3SUM-hard; for the second, we show that computing the coarseness with k boxes is NP-hard, when k is part of the input. For k fixed, we show that the coarseness can be computed in O(n2k−1) time and propose more efficient algorithms for k=2,3,4.

On Resilient Graph Spanners

We introduce and investigate a new notion of resilience in graph spanners. Let $$S$$S be a spanner of a weighted graph $$G$$G. Roughly speaking, we say that $$S$$S is resilient if all its point-to-point distances are resilient to edge failures. Namely, whenever any edge in $$G$$G fails, then as a consequence of this failure all distances do not degrade in $$S$$S substantially more than in $$G$$G (i.e., the relative distance increases in $$S$$S are very close to those in the underlying graph $$G$$G). In this paper we show that sparse resilient spanners exist, and that they can be computed efficiently.

An Inductive-style Procedure for Counting Monochromatic Simplexes of Symmetric Subdivisions with Applications to Distributed Computing

In the weak symmetry breaking (WSB) task, each of n+1 processes has to decide either 0 or 1 such that not all processes decide the same value. A WSB algorithm should be wait-free, namely, processes are asynchronous (there is no bound on their relative speeds), and potentially faulty (any proper subset may halt without warning). Also, it should be comparison based, namely, processes can only use comparison operations (<,>,=) on the values they read from the memory.Symmetric chromatic subdivisions of an n-simplex have been used to represent the executions of a distributed algorithm solving the WSB task. Informally, each simplex of such a subdivision corresponds to an execution of the algorithm. Each vertex is labeled with the local state of a process in the execution; it is colored with the process ID, and also with the binary value the process decides in the execution. The symmetry properties of such a complex come from the comparison based requirement of a WSB algorithm.Let C denote the number of monochromatic n-simplexes of such a subdivision, counted by orientation. Previous work has shown that C=1+∑i=0n−1(n+1i+1)ki for some coefficients ki∈Z. This characterization of C implies that the WSB task on n+1 processes is solvable if and only if n is such that the binomial coefficients are relatively prime, or equivalently, if and only if n is not a primer power.This paper presents an inductive style procedure that yields an alternative proof of the characterization of C. Roughly speaking, the proof consists in a procedure for modifying gradually the binary coloring of a symmetric chromatic subdivision, and computing the degree of the maps produced during the procedure.

Congruence conditions, parcels, and Tutte polynomials of graphs and matroids

Let G be a matrix and M(G) be the matroid defined by linear dependence on the set E of column vectors of G. Roughly speaking, a parcel is a subset of pairs (f,g) of functions defined on E to a suitable Abelian group A satisfying a coboundary condition (that the difference f−g is a flow over A of G) and a congruence condition (that an algebraic or combinatorial function of f and g, such as the sum of the size of the supports of f and g, satisfies some congruence condition). We prove several theorems of the form: a linear combination of sizes of parcels, with coefficients roots of unity, equals a multiple of an evaluation of the Tutte polynomial of M(G) at a point (u,v), usually with complex coordinates, satisfying (u−1)(v−1)=|A|.

On the complexity of Schmüdgen's Positivstellensatz

Schmüdgen's Positivstellensatz roughly states that a polynomial f positive on a compact basic closed semialgebraic subset S of R n can be written as a sum of polynomials which are non-negative on S for certain obvious reasons. However, in general, you have to allow the degree of the summands to exceed largely the degree of f. Phenomena of this type are one of the main problems in the recently popular approximation of non-convex polynomial optimization problems by semidefinite programs. Prestel (Springer Monographs in Mathematics, Springer, Berlin, 2001) proved that there exists a bound on the degree of the summands computable from the following three parameters: The exact description of S, the degree of f and a measure of how close f is to having a zero on S. Roughly speaking, we make explicit the dependence on the second and third parameter. In doing so, the third parameter enters the bound only polynomially.

ON THE EXISTENCE OF A SMOOTH CLOSED GEODESIC IN A CERTAIN MANIFOLD WITH NONSMOOTH BOUNDARY

where M is a smooth (of class C∞) finite-dimensional manifold regularly embedded in the space R, {f(·, u)} is a family of smooth (of class C∞) vector fields on M smoothly depending on the control parameter u, and U is a convex compact polyhedron in R. Roughly speaking, our main theorem (Theorem 1.1) here asserts that if system (0.1) satisfies the conditions of the existence theorem proved by the author in [8] and the vectorgram

A Classification of the Principal Nilpotent Pairs in Simple Lie Algebras and Related Problems

Let g be a semisimple Lie algebra over an algebraically closed field K of characteristic zero and G be its adjoint group. The notion of a principal nilpotent pair is a double counterpart of the notion of a regular (= principal) nilpotent element in g. Roughly speaking, a principal nilpotent pair e = (e1, e2) consists of two commuting elements in g that can independently be contracted to the origin and such that their simultaneous centralizer has the minimal possible dimension, that is, rkg. The definition and the basic results are due to V. Ginzburg [3]. He showed that the theory of principal nilpotent pairs yields a refinement of well-known results by B. Kostant on regular nilpotent elements in g and has interesting applications to representation theory. In particular, he proved that the number of G-orbits of principal nilpotent pairs is finite and gave a classification for g = sl(V). Trying to achieve a greater generality, Ginzburg also introduced a wider class of distinguished nilpotent pairs and, again, classified them for sl(V). (The precise definitions for all notions related to nilpotent pairs are found in §1.)

Topological conjugacy of singularities of vector fields

ROUGHLY speaking, a smooth vector field is finitely determined for topological conjugacy at a singularity if its conjugacy class contains some finite Taylor approximation. Finite determinacy is one criterion for the existence of conjugacies between smooth planar vector fields. This criterion can fail at a singularity only if the vector field is either P-flat or not sufficiently smooth. In both situations we construct specific examples of singularities. not rotation points, that are topologically equivalent but not conjugate. In addition, we determine a useful necessary and sufficient condition for the conjugacy of a large class of singularities. This allows for a simple characterization of the resulting conjugacy classes.

A mountain pass theorem

143 In other words, the mountain separating 0 and e may even be assumed of “zero” altitude and the conclusion is still true; moreover (as we shall show) if c = a the critical point can be chosen with llx,,ll = R. It is interesting to ask whether this extension of the Ambrosetti- Rabinowitz theorem remains true in the infinite-dimensional case. We prove here that if (1) is strengthened a little, to the form (1’) there exist real numbers u, r, R such that 0 a for every x E A := {X E X: Y a. Moreover, if c = a the critical point can be chosen with r < I/x0/I < R. Roughly speaking, in brief, the mountain pass theorem continues to hold for a mountain of zero altitude, provided it also has non-zero thickness; in addition, if c = a, the “pass” itself occurs precisely on the mountain-i.e., satisfying r < llroll < R. There are two interesting and immediate corollaries. The first one says that a C’ function which has IWO local minimum points also has a third critical point. The second states that a u-periodic C’ function with a local minimum e has a critical point x,, # e + ku, k = 0, + 1, +2 ,.... The precise statements of the above results will be given in the next sec- tions. The proof depends upon a lemma of Clark [3], formulated here in a version suitable to our purpose. In Section 3 some applications are presen- ted for the forced pendulum equation. 2.

Anthropomorphism

Anthropomorphism

Bounded approximation by polynomials

Abstract : This paper presents a complete solution to the following problem: if G is an arbitrary bounded open set in the complex plane, characterize those functions in G that can be obtained as the bounded pointwise limits of polynomials in G. Roughly speaking, the answer is that a function is such a limit if and only if it has a bounded analytic continuation throughout a certain bounded open set G* that contains G. This set G* is the inside of the 'outer boundary' of G. More precisely, if G is a bounded open set and if H is the unbounded component of the complement of G- (the closure of G), then G* denotes the complement of H-.

SEDIMENTATION CONSTANT-MOLECULAR WEIGHT RELATION OF RIBOSOMES.

ROUGHLY speaking, RNA of high molecular weight could be regarded as a polymer composed of nearly the same units as in the synthetic high polymer, if the minor differences in the structure of its base components are assumed to be negligible and hence the relation such as [η] = KMα would be expected between its intrinsic viscosity [η] and its molecular weight, M, which has been well known for the synthetic high polymers since the pioneer works of Staudinger. Indeed, Kurland1 proved it on the ribosomal RNA, reporting that the value of α is nearly 0.5. Summarizing the data reported hitherto, Iso2 also pointed out the following relation: is applicable to the RNA of the high molecular weight, where S0 is the sedimentation constant in Svedberg units and M the molecular weight. The value of the exponent in these formations is very close to those calculated theoretically on the model of spherical molecules of matted chains without free drain, a fact which provides us with an indirect support for the view of Hall and Doty3 that RNA molecules could be regarded as an ellipsoid of revolution hydrodynamically.

TESTING FOR SERIAL CORRELATION IN SYSTEMS OF SIMULTANEOUS REGRESSION EQUATIONS

where y is a vector of jointly dependent variables, x is a vector of predetermined or independent variables, e is a vector of errors, and A and B are matrices of unknown parameters. It is usual to assume that A is square and non-singular, that the e's are random variables with zero means and constant variance matrix, and that the x's are either constants or random variables distributed independently of the e's. Given the model (1) we usually require to estimate the elements of A and B from a sample of n observations of y and x and to make confidence statements about the estimates. In some formulations certain of the x's coincide with lagged values of the y's. The theory becomes much more complicated in such cases, and we shall not consider them except to point out that the results obtained later in the paper, which are exact for the model specified above, may be expected to hold approximately for models containing lagged dependent variables. The specification (1) is a natural generalization of the ordinary linear regression model and calls for similar methods of analysis. However, a numbor of new difficulties present themselves, among which is the identification problem, that is the question whether the parameter values can be inferred from a complete knowledge of the distribution of the observations. It turns out that this is possible under reasonable assumptions only when certain restrictions are obeyed by the elements of A and B. Roughly speaking, when the restrictions are just sufficient to enable the parameters of a particular equation to be estimated the equation is said to be just-identified, and when the restrictions are more than sufficient the equation is said to be over-identified. More detailed explanations are given by Girshick & Haavelmo (1947) and Koopmans and others (1950). When an equation is identified its parameters can be consistently estimated by the reduced-form method or the limited-information method, a treatment of which has been given by Anderson & Rubin (1949, 1950). Although some writers take the phrases as equivalent, in the present paper we shall use the phrase reduced-form method to refer only to estimation in the just-identified case while limited-information method will be taken to refer only to estimation in the over-identified case. These methods have the advantage that the parameters of the particular equation are estimated independently of the remaining parameters of the system. This is convenient since in many investigations we are not interested in the estimation of all the parameters of the system but only of those in a specific equation. The methods involve a possible loss of efficiency since the restrictions on the elements of A and B other than those in the equation under study are ignored in the