Duality theorems for a new class of multitime multiobjective variational problems

In this work, we consider a new class of multitime multiobjective variational problems of minimizing a vector of functionals of curvilinear integral type. Based on the normal efficiency conditions for multitime multiobjective variational problems, we study duals of Mond-Weir type, generalized Mond-Weir-Zalmai type and under some assumptions of (??, b)-quasiinvexity, duality theorems are stated. We give weak duality theorems, proving that the value of the objective function of the primal cannot exceed the value of the dual. Moreover, we study the connection between values of the objective functions of the primal and dual programs, in direct and converse duality theorems. While the results in §1 and §2 are introductory in nature, to the best of our knowledge, the results in §3 are new and they have not been reported in literature.     

Asymptotic inference for AR(1) processes with (nonnormal) stable errors

This is the first of several papers in which we consider problems related to the asymptotic distribution of the least squares estimate of the parameter γ in theAR(1) model $$X_k = \gamma X_{k - 1} + \varepsilon _k , k = 1,...,n,$$ where ε_k are independent identically distributed (i.i.d.) random variables in the domain of attraction of a stable law. In §1 we give a summary in the case ε_k is in the domain of attraction of the normal distribution. In §2 we consider errors in the domain of attraction of a (nonnormal) stable distribution. In §3 we prove a result in the case of the completely asymmetric stable distribution with α=β=1.     

On pure uniform holomorphy in spaces of holomorphic germs

If E is a complex (DFC)-space (see § 2), we show that E leads to pure uniform holomorphy (see §2) if and only if its Fréchet dual space E′ is separable (see Theorem 1, where these two conditions have other eight equivalent ones). By using a theorem of Mujica (see §4), we consider the (DFC)-space K(K) of germs around K of holomorphic C-valued functions where K is a nonvoid compact subset of a complex metrizable locally convex space E, and ?(K) is endowed with the topology ?₀ obtained as an inductive limit of compact-open topologies (see §4). Not only Theorem 1 applies to ?(K), with E replaced by ?(K) in its statement, but also ?(K) leads to pure uniform holomorphy if and only if E is separable (see Theorem 2).     

Initial-Boundary Value Problem for a Class of Linear Relaxation Systems in Arbitrary Space Dimensions

In this paper, we consider the characteristic initial-boundary value problem (IBVP) for the multi-dimensional Jin-Xin relaxation model in a half-space with arbitrary space dimension n?2. As in the one-dimensional case (n=1, see (J. Differential Equations, 167 (2000), 388-437), our main interest is on the precise structural stability conditions on the relaxation system, particularly the formulation of boundary conditions, such that the relaxation IBVP is stiffly well posed, that is, uniformly well posed independent of the relaxation parameter ε>0, and the solution of the relaxation IBVP converges, as ε→0, to that of the corresponding limiting equilibrium system, except for a sharp transition layer near the boundary. Our main result can be roughly stated as Stiff Kreiss Condition=Uniform Kreiss Condition for the relaxation IBVP we consider in this paper, which is in sharp contrast to the one-dimensional case (Z. Xin and W.-Q. Xu, J. Differential Equations, 167 (2000), 388-437). More precisely, we show that the Uniform Kreiss Condition (which is necessary and sufficient for the well posedness of the relaxation IBVP for each fixed ε), together with the subcharacteristic condition (which is necessary and sufficient for the stiff well posedness of the corresponding Cauchy problem), also guarantees the stiff well posedness of our relaxation IBVP and the asymptotic convergence to the corresponding equilibrium system in the limit of small relaxation rate. Optimal convergence rates are obtained and various boundary layer behaviors are also rigorously justified.     

Bolzano’s Argument for the Existence of Substances: a Formalization with Two Types of Predication

The topic of our analysis is the argument for the existence of substances given by Bernard Bolzano in Athanasia (1827), where he essentially employs two ontological categories: substance and adherence. Bolzano considers the real and conditioned Inbegriff of all adherences, which are wirklich and nicht selbst bestehen. He claims that the formed collection is dependent on something external and non-adherential, which therefore is a substance. Bolzano’s argumentation turns out to be structurally similar to his argument for the existence of God from Lehrbuch der Religionswissenschaft (1834), but in each of these reasonings, we find different plausible interpretations of the key concept “Inbegriff”. The latter argumentation refers to the mereological totality of existentially conditioned objects. We propose the explication of the Bolzanian Inbegriff of all adherences using two types of predication: we consider its extension as composed of certain intensional counterparts of adherences. In our approach, we use a fragment of the theory of abstract objects formulated by E. Zalta (1983), describing two different relations between individuals and properties: extensional exemplification and intensional encoding. We put our reconstruction in a wider context of Bolzano’s ontology, formulating the needed axioms with two primitive predicates of second order ... is an adherence, ... is conditioned by something real as well as the conditionally introduced first order predicate constant \(\mathfrak {In}\) for Inbegriff of all adherential ideas. Finally, we sketch a model for our theory.     

Necklace rings and logarithmic functions

In this paper, we develop the theory of the necklace ring and the logarithmic function. Regarding the necklace ring, we introduce the necklace ring functor Nr from the category of special λ-rings into the category of special λ-rings and then study the associated Adams operators. As far as the logarithmic function is concerned, we generalize the results in Bryant's paper [Free Lie algebras and formal power series, J. Algebra 253(1) (2002) 167-188] to the case of graded Lie (super)algebras with a group action by applying the Euler-Poincaré principle.     

On the Modular Figure and Ford-Spheres

In §1 we study Figure 1 for variable h. In §2 and §3 this is combined with Ford-circles (Figure 2) and the choice h = 0 (Figure 3). In §4 a limit is evaluated. In §5 we treat Figure 5 (“Ford-circles inside the unit-circle”) and a technical application. In §6 contains various modular figures (Figures 7, 8, 9). §7 treats synchronous rotation of Figure 1 and Eoing from R² to R³; the rotating upper half-plane of C corresponds then to the unit-sphere. In §8 a sphere-packing on C is introduced. Finally in §9 a sphere-packing inside the unit-sphere arises.     

Coincident Entities and Question-Begging Predicates: an Issue in Meta-Ontology

Meta-ontology (in van Inwagen's sense) concerns the methodology of ontology, and a controversial meta-ontological issue is to what extent ontology can rely on linguistic analysis while establishing the furniture of the world. This paper discusses an argument advanced by some ontologists (I call them unifiers) against supporters of or coincident entities (I call them multipliers) and its meta-ontological import. Multipliers resort to Leibniz's Law to establish that spatiotemporally coincident entities a and b are distinct, by pointing at a predicate F() made true by a and false by b. Unifiers try to put multipliers in front of a dilemma: in attempting to introduce metaphysical differences on the basis of semantic distinctions, multipliers either (a) rest on a fallacy of verbalism, entailed by a trade-off between a de dicto and a de re reading of modal claims, or (b) beg the question against unifiers by having to assume the distinction between a and b beforehand. I shall rise a tu quoque, showing that unifiers couldn't even distinguish material objects (or events) from the spatiotemporal regions they occupy unless they also resorted to linguistic distinctions. Their methodological aim to emancipate themselves from linguistic analysis in ontological businesses is therefore problematic.     

Strong pro-fibrations and ANR objects in pro-categories

The notions of pro-fibration and approximate pro-fibration for morphisms in the pro-category pro-Top of topological spaces were introduced by S. Mardeši? and T.B. Rushing. In this paper we introduce the notion of strong pro-fibration, which is a pro-fibration with some additional property, and the notion of ANR object in pro-Top, which is approximately an ANR-system, and we consider the full subcategory ANR of pro-Top whose objects are ANR objects. We prove that the category ANR satisfies most of the axioms for fibration category in the sense of H.J. Baues if fibrations are strong pro-fibrations and weak equivalences are morphisms inducing isomorphisms in the pro-homotopy category pro-H(Top) of topological spaces. We give various applications. First of all, we prove that every shape morphism is represented by a strong pro-fibration. Secondly, the fibre of a strong pro-fibration is well defined in the category ANR, and we obtain an isomorphism between the pro-homotopy groups of the base and total systems of a strong pro-fibration, and hence obtain the pro-homotopy sequence of a strong pro-fibration. Finally, we also show that there is a homotopy decomposition in the category ANR.     

Nonparametric Estimation of the Finite-Time Survival Probability with Zero Initial Capital in the Classical Risk Model

In a classical risk model with zero initial capital and unknown claim-size distribution, we consider the statistical problem of estimating uniformly in t the (unknown) finite-time survival probability ?? ₀(t) at time t, given a sample of claim sizes. We construct an empirical estimator of the function ?? ₀(·) based on the sample of claim sizes, and using a functional approach we establish asymptotic statistical properties of our estimator with respect to supremum norm. We also consider numerical evaluation of finite-time survival probabilities and their empirical counterparts using the fast Fourier transform algorithm, and we carry out small-scale simulation studies of the behaviour of our estimator.     

On subgroups of finite <Emphasis Type="Italic">p</Emphasis>-groups

In §2, we prove that if a 2-group G and all its nonabelian maximal sub-groups are two-generator, then G is either metacyclic or minimal non-abelian. In §3, we consider a similar question for p > 2. In §4 the 2-groups all of whose minimal nonabelian subgroups have order 16 and a cyclic subgroup of index 2, are classified. It is proved, in §5, that if G is a nonmetacyclic two-generator 2-group and A, B, C are all its maximal subgroups with d(A) ≤ d(B) ≤ d(C), then d(C) = 3 and either d(A) = d(B) = 3 (this occurs if and only if G/G′ has no cyclic subgroup of index 2) or else d(A) = d(B) = 2. Some information on the last case is obtained in Theorem 5.3.     

Improved Results on Geometric Hitting Set Problems

We consider the problem of computing minimum geometric hitting sets in which, given a set of geometric objects and a set of points, the goal is to compute the smallest subset of points that hit all geometric objects. The problem is known to be strongly NP-hard even for simple geometric objects like unit disks in the plane. Therefore, unless P = NP, it is not possible to get Fully Polynomial Time Approximation Algorithms (FPTAS) for such problems. We give the first PTAS for this problem when the geometric objects are half-spaces in ?³ and when they are an r-admissible set regions in the plane (this includes pseudo-disks as they are 2-admissible). Quite surprisingly, our algorithm is a very simple local-search algorithm which iterates over local improvements only.     

Locating an axis-parallel rectangle on a Manhattan plane

In this paper we consider the problem of locating an axis-parallel rectangle in the plane such that the sum of distances between the rectangle and a finite point set is minimized, where the distance is measured by the Manhattan norm ? ₁. In this way we solve an extension of the Weber problem to extensive facility location. As a model, our problem is appropriate for position sensing of rectangular objects.     

on O-simple semirings,semigroup semirings,and two kinds of division semirings

In this paper we consider O-simple semirings S, where O denotes the multiplicative zero of S, which may be in particular the additive neutral o of S at the same time. In this context we give some statements on matrix semirings and introduce contracted semigroup semirings in §3, a matter of interest of its own. We further use our results to compare the usual concept of division semirings with a new one introduced in [18], and we show that a corresponding theorem in [18] is in general only valid for division semirings in the usual meaning. Dedicated to E.S. Lyapin on his 70th birthday     

Left ideals of polynomial rings in several indeterminates

We consider maximal left ideals L of the polynomial ring R[λ₁, …, λ_n], for R noncommutative. In §1 we reprove and generalize Resco's result that any maximal left ideal L is generated by ≤ n elements whenever R is simple Artinian, and obtain more precise information about the generators when R satisfies a PI. In many instances, fewer than n generators suffice; this is considered in §3, by means of various examples. In §2 we see by a straightforward argument that L has bounded height as a prime left ideal whenever R is a simple Pl-ring, but this does not- hold in general for R simple Artinian.     

An analytic version of the De Rham theorem

LetV be a real analytic paracompact variety; in §1 of this paper we prove that: $$H^q (V,R) \approx \frac{{closed analytic differentiable q - forms on V}}{{exact analytic differentiable q - forms on V}}$$ Then we prove that the closed (exact) analytic differentiableq-forms onV are dense, in the Whitney topology, in the set of closed (exact) differentiableq-forms onV. We also consider the problem of extending closed (exact) analytic differentiableq-forms, defined on a subvarietyX ofV, to closed (exact) analytic forms defined onV.     

Constrained principal component analysis of standardized data for biplots with unit-length variable vectors

Principal component analysis (PCA) of an objects ×  variables data matrix is used for obtaining a low-dimensional biplot configuration, where data are approximated by the inner products of the vectors corresponding to objects and variables. Borg and Groenen (Modern multidimensional scaling. Springer, New York, 1997) have suggested another biplot procedure which uses a technique for approximating data by projections of object vectors on variable vectors. This technique is formulated as constraining the variable vectors in PCA to be of unit length and can be called unit-length vector analysis (UVA). However, an algorithm for UVA has not yet been developed. In this paper, we present such an algorithm, discuss the properties of UVA solutions, and demonstrate the advantage of UVA in biplots for standardized data with homogeneous variances among variables. The advantage of UVA-based biplots is that the projections of object vectors onto variable vectors express the approximation of data in an easy way, while in PCA-based biplots we must consider not only the projections, but also the lengths of variable vectors in order to visualize approximations.     

Statistical lacunary summability and a Korovkin type approximation theorem

In this paper, we introduce statistical lacunary summability and strongly ?? _q -convergence (0 < q < ??) and establish some relations between lacunary statistical convergence, statistical lacunary summability, and strongly ?? _q -convergence. We further apply our new notion of summability to prove a Korovkin type approximation theorem.     

Relative rigid objects in extriangulated categories

In this paper, we study a close relationship between relative cluster tilting theory in extriangulated categories and τ-tilting theory in module categories. Our main results show that relative rigid objects are in bijection with τ-rigid pairs, and also relative maximal rigid objects with support τ-tilting pairs under some assumptions. These results generalize the work by Adachi-Iyama-Reiten, Yang-Zhu and Fu-Geng-Liu. In addition, we introduce mutation of relative maximal rigid objects and show that any basic relative almost maximal rigid object has exactly two non-isomorphic indecomposable complements. All results highlight new phenomena when they applied to exact categories.