Counter Example Research Articles

Existence, uniqueness and ergodicity for the centered Fleming–Viot process

Motivated by questions of ergodicity for shift invariant Fleming–Viot process, we consider the centered Fleming–Viot process Ztt⩾0 defined by Zt≔τ−id,Yt♯Yt, where Ytt⩾0 is the original Fleming–Viot process. Our goal is to characterize the centered Fleming–Viot process with a martingale problem. To establish the existence of a solution to this martingale problem, we exploit the original Fleming–Viot martingale problem and asymptotic expansions. The proof of uniqueness is based on a weakened version of the duality method, allowing us to prove uniqueness for initial conditions admitting finite moments. We also provide counter examples showing that our approach based on the duality method cannot be expected to give uniqueness for more general initial conditions. Finally, we establish ergodicity properties with exponential convergence in total variation for the centered Fleming–Viot process and characterize the invariant measure.

On Some New Uncertain Spaces using Generalized Difference Operator

It was C. You, who has given the concept of uncertainty theory in 2009. The scenario of this paper is to define a new notion of spaces using generalized ∆- operator and the uncertainty theory. Also, certain basic structures will be given. Moreover, inclusion relations and their counter examples will be given.

On asymptotic dark energy in string theory

We examine bounds on accelerated expansion in asymptotic regions of the moduli space in string theory compactifications to four spacetime dimensions. While there are conjectures that forbid or constrain accelerated expansion in such asymptotic regions, potential counter examples have been discussed recently in the literature. We check whether such counter examples can arise in explicit string theory constructions, focusing in particular on non-geometric compactifications of type IIB string theory that have no Kähler moduli. We find no violation of the Strong Asymptotic dS Conjecture and thus provide support for the absence of accelerated expansion in asymptotic regions of a barely explored corner of the string landscape. Moreover, working in a simplified setting, we point out a new mechanism for potentially connecting the Sharpened Distance Conjecture and the Strong Asymptotic dS Conjecture. If this argument could be generalized, it would mean that the Sharpened Distance Conjecture is implied by the Strong Asymptotic dS Conjecture, and that their exponential factors are naturally related by a factor of 2.

Existence of a variational principle for PDEs with symmetries and current conservation

It is well-known from Noether's theorem that symmetries of a variational functional lead to conservation laws of the corresponding Euler-Lagrange equation. We reverse this statement and prove that a differential equation, which satisfies sufficiently many symmetries and corresponding conservation laws, leads to a variational functional, whose Euler-Lagrange equation is the given differential equation. Sufficiently many symmetries means that the set of symmetry vector fields satisfy span{Vp:V∈Sym}=TpE, that is, they span the tangent space TpE at each point p of a fiber bundle E, which describes the dependent- and independent coordinates. Higher order coordinates are described by the jet bundle JkE and we require no span-assumptions on TJkE. Our main theorem states that Noether's theorem can be reversed in this sense for second order differential equations, or more precisely, for so-called second order source forms on J2E, which are required to write the differential equation as a weak formulation (every Euler-Lagrange equation is derived from a first variation, that is, from a weak formulation). Counter examples show that our theorem is sharp.

Analysis of nonconforming IFE methods and a new scheme for elliptic interface problems

In this paper, an important discovery has been found for nonconforming immersed finite element (IFE) methods using the integral values on edges as degrees of freedom for solving elliptic interface problems. We show that those IFE methods without penalties are not guaranteed to converge optimally if the tangential derivative of the exact solution and the jump of the coefficient are not zero on the interface. A nontrivial counter example is also provided to support our theoretical analysis. To recover the optimal convergence rates, we develop a new nonconforming IFE method with additional terms locally on interface edges. The new method is parameter-free which removes the limitation of the conventional partially penalized IFE method. We show the IFE basis functions are unisolvent on arbitrary triangles which is not considered in the literature. Furthermore, different from multipoint Taylor expansions, we derive the optimal approximation capabilities of both the Crouzeix–Raviart and the rotated-Q1 IFE spaces via a unified approach which can handle the case of variable coefficients easily. Finally, optimal error estimates in both H1- and L2-norms are proved and confirmed with numerical experiments.

A short note on “A note on single-machine scheduling with job-dependent learning effects”

Mosheiov & Sidney [5] showed that the single-machine makespan minimization problem with job-dependent learning effects can be formulated as an assignment problem and solved in O(n3) time. Later on, Koulamas [4] showed that the scheduling problem can be solved in O(nlogn) time by sequencing the jobs according to the shortest processing time (SPT) rule if the job-dependent learning rate are correlated with the level of sophistication of the jobs and bounded from below. The optimality of the SPT job sequence still holds when job-dependent learning rate are inversely correlated with the level of sophistication of the jobs and bounded from above. The main results in the paper are correct, however, there is an error in Corollary 1 and the proofs of Proposition 1 and Corollary 1 are not complete. In this note, a counter example for the latter case is given first and then a modified corollary is provided. Next, a lemma is provided to complete the proofs of Proposition 1 and Corollary 1. Finally, a simple algorithm is developed to solve the latter case in O(n2) time.

Various results on some asymmetric types of density

The structures of symmetric connectedness and dually, antisymmetric connectedness were described and studied before, especially in terms of graph theory as the corresponding counterparts of the connectedness of a graph and the connectedness of its complementary graph. By taking into consideration the deficiencies of topological density in the context of symmetric and antisymmetric connectedness, two special kinds of density in the theory of non-metric $T_0$-quasi-metrics were introduced in the previous studies under the names symmetric density and antisymmetric density. In this paper, some crucial and useful properties of these two types of density are investigated with the help of the major results and (counter)examples peculiar to the asymmetric environment. Besides these, many further observations about the structures of symmetric and antisymmetric-density are dealt with, especially in the sense of their combinations such as products and unions through various theorems in the context of $T_0$-quasi-metrics. Also, we examine the question of under what kind of quasi-metric mapping these structures will be preserved.

A Note on Controllability of Directed Networked MIMO Systems With Heterogeneous Dynamics

In Xiang et al. [6], a necessary condition for the controllability of a special type of heterogeneous networked system is derived. Theorem 3 in Xiang et al. states that the observability of each node is necessary for the controllability of a heterogeneous networked system satisfying (i) similar state matrices, (ii) output matrices are scalar multiples of one another and (iii) rank of the input matrix of the networked system is less than the number of nodes in the system. In this short article, we have given a counter example to show that this claim is not true in general. Further, we have given a necessary condition for the controllability of a heterogeneous networked system with above mentioned properties. The result obtained is demonstrated through examples.

A Rademacher type theorem for Hamiltonians H(x, p) and an application to absolute minimizers

We establish a Rademacher type theorem involving Hamiltonians H(x, p) under very weak conditions in both of Euclidean and Carnot-Carathéodory spaces. In particular, H(x, p) is assumed to be only measurable in the variable x, and to be quasiconvex and lower-semicontinuous in the variable p. Without the lower-semicontinuity in the variable p, we provide a counter example showing the failure of such a Rademacher type theorem. Moreover, by applying such a Rademacher type theorem we build up an existence result of absolute minimizers for the corresponding L^infty -functional. These improve or extend several known results in the literature.

SOMEWHAT gp-CONTINUOUS AND SOMEWHAT gp-IRRESOLUTE FUNCTIONS

The main object of this paper is to study the basic properties of somewhat gp-continuous functions using the concept of gp-open sets in topological spaces. Also, the concept of somewhat gp-open functions and somewhat gpirresolute functions with some counter examples are explained in this paper. Further, the author establishes the relationship between the new classes of functions with other classes of functions by giving examples, counterexamples, properties and characterizations.

Transfeminine letter clubs, community care and the radical politics of the erotic

This essay examines trans feminine community-building and care in the context of desire and the erotic as forms of political praxis. The theoretical core of the text builds on Audre Lorde’s broad, poetic use of the erotic as well as José Muñoz’s aesthetic and utopian notion of queer future. I argue that Lorde and Muñoz provide important tools for theorizing the erotic as a politically subversive and utopian form of care in the contemporary political setting of rising anti-genderism, moral conservativism and neo-fascism. The essay begins though a brief, historical examination of the ways in which transgender sexuality, desire and the erotic have come to be a neglected and contested topic in both academic and clinical research, despite what Michel Foucault defines as a ‘discursive explosion’ on the truth about sex and sexuality in modernity. I will then point out historical counter examples of transfeminine, transvestite and cross-dresser community-building in which care is intimately linked to the erotic in Lorde’s sense of the term. Finally, I argue that trans desire and the erotic should be understood as radically political in Muñoz’s sense of utopian imagining of a future yet to be made.

A counter example on a Borsuk conjecture

The study of shape restrictions of subsets of Rd has several applications in many areas, being convexity, r-convexity, and positive reach, some of the most famous, and typically imposed in set estimation. The following problem was attributed to K. Borsuk, by J. Perkal in 1956:find an r-convex set which is not locally contractible. Stated in that way is trivial to find such a set. However, if we ask the set to be equal to the closure of its interior (a condition fulfilled for instance if the set is the support of a probability distribution absolutely continuous with respect to the d-dimensional Lebesgue measure), the problem is much more difficult. We present a counter example of a not locally contractible set, which is r-convex. This also proves that the class of supports with positive reach of absolutely continuous distributions includes strictly the class ofr-convex supports of absolutely continuous distributions.

A theoretical approach on intuitionistic Fuzzy Hausdorff space

The object of this paper is to introduce a new definition for intuitionistic fuzzy Hausdorff space (IFHS). We investigate some of its characterizations and discuss it with some necessary counter examples. In addition, we compared the new notion with the existing notions. Finally we point out the significance of Hausdorffness in digital image processing.

High-energy properties of the graviton scattering in quadratic gravity

We obtain the matter-graviton scattering amplitude in the gravitational theory of quadratic curvature, which has {R}_{mu nu}^2 term in the action. Unitarity bound is not satisfied because of the existence of negative norm states, while an analog of unitarity bound for S-matrix unitarity holds due to the cancelation among the positive norm states and negative norm ones in the unitarity summation in the optical theorem. The violation of unitarity bound is a counter example of Llewellyn Smith’s conjecture on the relation between tree-level unitarity and renormalizability. We have recently proposed a new conjecture that an analog of the unitarity bound for S-matrix unitarity gives the equivalent conditions to those for renormalizability. We show that the gravitational theory of quadratic curvature is a nontrivial example consistent with our conjecture.

Argumentative strategies against scientism: an overview

ABSTRACT This paper presents and discusses various strategies that have been wielded against scientism, roughly the claim that only science can give us knowledge. The strategies identified are: (1) the counter example strategy, (2) the denying of claimed entailments of science strategy, (3) the self-undermining strategy, (4) the presupposition strategy, and (5) the limits of science strategy. In addition, two proposals are discussed that aim to recast the debate about scientism in a way that renders these strategies obsolete. It is argued that these proposals are misguided.

Statistical calibration for infinite many future values in linear regression: simultaneous or pointwise tolerance intervals or what else?

AbstractStatistical calibration using regression is a useful statistical tool with many applications. For confidence sets for x-values associated with infinitely many future y-values, there is a consensus in the statistical literature that the confidence sets constructed should guarantee a key property. While it is well known that the confidence sets based on the simultaneous tolerance intervals (STIs) guarantee this key property conservatively, it is desirable to construct confidence sets that satisfy this property exactly. Also, there is a misconception that the confidence sets based on the pointwise tolerance intervals (PTIs) also guarantee this property. This paper constructs the weighted simultaneous tolerance intervals (WSTIs) so that the confidence sets based on the WSTIs satisfy this property exactly if the future observations have the x-values distributed according to a known specific distribution F(⋅). Through the lens of the WSTIs, convincing counter examples are also provided to demonstrate that the confidence sets based on the PTIs do not guarantee the key property in general and so should not be used. The WSTIs have been applied to real data examples to show that the WSTIs can produce more accurate calibration intervals than STIs and PTIs.

The polynomial reconstruction problem: The first 50 years

The problem of reconstructing the characteristic polynomial of a graph of order at least 3 from the collection of characteristic polynomials of its vertex-deleted subgraphs was posed by Cvetković in 1973 as a spectral counter part to the well-known Ulam's reconstruction conjecture. Over the last 50 years, this problem has received notable attention, many positive results have been obtained, but in the general case the problem is still unresolved. In particular, no counter example is found in literature. In this expository paper we survey classical and some more recent results concerning the polynomial reconstruction problem, discuss some related problems, variations and generalizations.

SAT-Reach : A Bounded Model Checker for Affine Hybrid Systems

Bounded model checking (BMC) is well-known to be undecidable even for simple hybrid systems. Existing work targeted for a wide class of non-linear hybrid systems reduces the BMC problem to the satisfiability problem of an satisfiability modulo theory formula encoding the hybrid system dynamics. Consequently, the satisfiability of the formula is deduced with a δ-decision procedure. However, the encoded formula can be complex for large automaton and for deep exploration causing the decision procedure to be inefficient. Additionally, a generalized decision procedure can be inefficient for hybrid systems with simple dynamics. In this article, we propose a BMC algorithm built upon the foundation of the counter example guided abstraction refinement (CEGAR) technique and targeted for hybrid systems with piecewise affine dynamics, modeled as a hybrid automaton. In particular, our algorithm begins by searching an abstract counterexample in the discrete state-space of the automaton. We check whether a discovered abstract counterexample is spurious or real by a two-tier refinement of the state-space guided by the abstract counterexample. The primary refinement is through symbolic reachability analysis and the following refinement is via a search of a real counterexample by the trajectory splicing method, guided in turn by the outcome of reachability analysis. We show that our algorithm reaps the benefits of the CEGAR technique by directing the exploration in the regions of interest and pruning search space that is irrelevant to the property under consideration. In addition, an optimization by memoizing the computed symbolic states during reachability analysis has been proposed for efficiency. The proposed algorithm is implemented in the tool SAT-Reach, and we compare its performance with dReach , XSpeed , Flow* , SpaceEx, and a pattern database heuristic-guided search algorithm. Experiments demonstrate the efficacy of our algorithm.

Addendum to: “The Friedrichs angle and alternating projections in Hilbert C⁎-modules” [J. Math. Anal. Appl. 516 (2022) 126474

We show by (counter)example that the intersection of complemented submodules in a Hilbert C⁎-module is not necessarily complemented, answering the last open question from [1].

The rangeomorph Pectinifrons abyssalis: Hydrodynamic function at the dawn of animal life

Rangeomorphs are among the oldest putative eumetazoans known from the fossil record. Establishing how they fed is thus key to understanding the structure and function of the earliest animal ecosystems. Here, we use computational fluid dynamics to test hypothesized feeding modes for the fence-like rangeomorph Pectinifrons abyssalis, comparing this to the morphologically similar extant carnivorous sponge Chondrocladia lyra. Our results reveal complex patterns of flow around P.abyssalis unlike those previously reconstructed for any other Ediacaran taxon. Comparisons with C. lyra reveal substantial differences between the two organisms, suggesting they converged on a similar fence-like morphology for different functions. We argue that the flow patterns recovered for P.abyssalis do not support either a suspension feeding or osmotrophic feeding habit. Instead, our results indicate that rangeomorph fronds may represent organs adapted for gas exchange. If correct, this interpretation could require a dramatic reinterpretation of the oldest macroscopic animals.