Search Results (7)

We study ratio-induced mismatch cost functions of the form $c(s,o)=J\left({\iota }_S\left(s\right)/{\iota }_O\left(o\right)\right)$ built from positive scale maps ${\iota }_S:S\to {\mathbb{R}}_{>0}$ and ${\iota }_O:O\to {\mathbb{R}}_{>0}$ and a penalty $J:(0,\infty )\to [0,\infty )$. Assuming inversion symmetry, strict convexity, coercivity, normalization at 1, and a multiplicative d’Alembert identity, we show that $f\left(u\right):=1+J\left(e^u\right)$ is continuous and satisfies the additive d’Alembert equation; hence, by a classical classification theorem, there exists $a>0$ such that $J\left(x\right)=cosh(alogx)-1={\displaystyle \frac{1}{2}}\left(x^a+x^{-a}\right)-1, x>0$. We then analyze the associated argmin mapping over feasible scale sets: existence under explicit subspace-closedness assumptions, an explicit geometric-mean decision geometry for finite dictionaries with stability away from boundaries, exact compositionality for product models, and an optimal sequential mediation principle described by a geometric mean (or its log-space projection when infeasible). The paper is purely mathematical; any semantic interpretation is optional and external to theorems proved here. Full article

We develop the theory of Difference Lindelöf perfect functions. Through difference covers, we provide intrinsic characterizations; prove stability under composition, subspace restriction, and suitable products; and obtain preservation theorems. Under standard separation axioms, properties such as D-countable compactness, regularity, paracompactness, and the closedness of projections transfer along D-Lindelöf perfect maps. We also connect the framework to statistics. Uses include decision regions expressed as differences of open sets and parameter screening, with visualizations of countable subcovers and their pushforwards. The results point to practical countable cores for learning and inference and suggest extensions to bitopological and fuzzy contexts. Full article

This study advances the application of the generalized Halmos’ two projections theorem to idempotent operators on Hilbert $C^{*}$-modules through a comprehensive study of sums involving adjointable idempotents and their adjoints. We establish fundamental properties including the closedness, orthogonal complementability, Moore–Penrose inverses, and spectral norms of such sums. For arbitrary (not necessarily adjointable) idempotent operators that admit a decomposition into linear combinations or products of two idempotents, we derive explicit representations for all such decompositions. A numerical example is given to show how our main theorem allows for the decomposition of idempotent matrices into linear combinations of two idempotent matrices, and two concrete examples on Hilbert $C^{*}$-modules validate the theoretical significance of our framework. Full article

In this paper, we propose a unified concept encompassing generalizations of two types of families defined based on Levine’s notions of generalized closed sets and Maki’s $\mathsf{\Lambda }$ sets. The methods used in this investigation are described in my previous work, where a unified concept of general closedness is presented. From a methodology point of view, the present concept is symmetric to the previous. In generalizing open subsets, one can use the two methods. According to the first one, the family of Levine’s generalization is used as some base to build the family of closed subsets of the new topology. In the second method, the family of open subsets is extended, in the same way, as the family of closed subsets in the classic Levine’s method. The results obtained in this general conception easily extend and imply well-known theorems of this area of investigation. In the literature on this issue, many versions of generalizations of $\mathsf{\Lambda }$-sets have been investigated. The tools used in this paper enabled us to prove that there exist at most 10 generalizations of these types, and we show the relationships between them in the graph. As a result, it turns out that some generalizations investigated in the literature are trivial. Full article

In this work, we study iterative methods for the approximation of common attractive points of two widely more generalized hybrid mappings in Hilbert spaces and obtain weak and strong convergence theorems without assuming the closedness for the domain. A numerical example supporting our main result is also presented. As a consequence, our main results can be applied to solving a common fixed point problem. Full article

We are concerned with rigid analytic geometry in the general setting of Henselian fields K with separated analytic structure, whose theory was developed by Cluckers–Lipshitz–Robinson. It unifies earlier work and approaches of numerous mathematicians. Separated analytic structures admit reasonable relative quantifier elimination in a suitable analytic language. However, the rings of global analytic functions with two kinds of variables seem not to have good algebraic properties such as Noetherianity or excellence. Therefore, the usual global resolution of singularities from rigid analytic geometry is no longer at our disposal. Our main purpose is to give a definable version of the canonical desingularization algorithm (the hypersurface case) due to Bierstone–Milman so that both of these powerful tools are available in the realm of non-Archimedean analytic geometry at the same time. It will be carried out within a category of definable, strong analytic manifolds and maps, which is more flexible than that of affinoid varieties and maps. Strong analytic objects are those definable ones that remain analytic over all fields elementarily equivalent to K. This condition may be regarded as a kind of symmetry imposed on ordinary analytic objects. The strong analytic category makes it possible to apply a model-theoretic compactness argument in the absence of the ordinary topological compactness. On the other hand, our closedness theorem enables application of resolution of singularities to topological problems involving the topology induced by valuation. Eventually, these three results will be applied to such issues as the existence of definable retractions or extending continuous definable functions. The established results remain valid for strictly convergent analytic structures, whose classical examples are complete, rank one valued fields with the Tate algebras of strictly convergent power series. The earlier techniques and approaches to the purely topological versions of those issues cannot be carried over to the definable settings because, among others, non-Archimedean geometry over non-locally compact fields suffers from lack of definable Skolem functions. Full article

We show that the recent results on the Fundamental Theorem of Asset Pricing and the super-hedging theorem in the context of model uncertainty can be extended to the case in which the options available for static hedging (hedging options) are quoted with bid-ask spreads. In this set-up, we need to work with the notion of robust no-arbitrage which turns out to be equivalent to no-arbitrage under the additional assumption that hedging options with non-zero spread are non-redundant. A key result is the closedness of the set of attainable claims, which requires a new proof in our setting. Full article