Some Hadamard–Fejér Type Inequalities for LR-Convex Interval-Valued Functions

The purpose of this study is to introduce the new class of Hermite–Hadamard inequality for LR-convex interval-valued functions known as LR-interval Hermite–Hadamard inequality, by means of pseudo-order relation ( ≤p ). This order relation is defined on interval space. We have proved that if the interval-valued function is LR-convex then the inclusion relation “ ⊆ ” coincident to pseudo-order relation “ ≤p ” under some suitable conditions. Moreover, the interval Hermite–Hadamard–Fejér inequality is also derived for LR-convex interval-valued functions. These inequalities also generalize some new and known results. Useful examples that verify the applicability of the theory developed in this study are presented. The concepts and techniques of this paper may be a starting point for further research in this area.

Some Integral Inequalities for Generalized Convex Fuzzy-Interval-Valued Functions via Fuzzy Riemann Integrals

In this study, we introduce the new concept of $$h$$ h -convex fuzzy-interval-valued functions. Under the new concept, we present new versions of Hermite–Hadamard inequalities (H–H inequalities) are called fuzzy-interval Hermite–Hadamard type inequalities for $$h$$ h -convex fuzzy-interval-valued functions ($$h$$ h -convex FIVF) by means of fuzzy order relation. This fuzzy order relation is defined level wise through Kulisch–Miranker order relation defined on fuzzy-interval space. Fuzzy order relation and inclusion relation are two different concepts. With the help of fuzzy order relation, we also present some H–H type inequalities for the product of $$h$$ h -convex FIVFs. Moreover, we have also established strong relationship between Hermite–Hadamard–Fej´er (H–H–Fej´er) type inequality and $$h$$ h -convex FIVF. There are also some special cases presented that can be considered applications. There are useful examples provided to demonstrate the applicability of the concepts proposed in this study. This paper's thoughts and methodologies could serve as a springboard for more research in this field.

Consistent Pitch Height Forms: A commentary on Daniel Muzzulini's contribution Isaac Newton's Microtonal Approach to Just Intonation

This text revisits selected aspects of Muzzulini's article and reformulates them on the basis of a three-dimensional interval space E and its dual E*. The pitch height of just intonation is conceived as an element h of the dual space. From octave-fifth-third coordinates it becomes transformed into chromatic coordinates. The dual chromatic basis is spanned by the duals a* of a minor second a and the duals b* and c* of two kinds of augmented primes b and c. Then for every natural number n a modified pitch height form hn is derived from h by augmenting its coordinates with the factor n, followed by rounding to nearest integers. Of particular interest are the octave-consitent forms hn mapping the octave to the value n. The three forms hn for n = 612, 118, 53 (yielding smallest deviations from the respective values of n h) form the Muzzulini basis of E*. The respective transformation matrix T* between the coordinate representations of linear forms in the Muzzulini basis and the dual chromatic basis is unimodular and a Pisot matrix with the dominant eigen-co-vector very close to h. Certain selections of the linear forms hn are displayed in Muzzuli coordinates as ball-like point clouds within a suitable cuboid containing the origin. As an open problem remains the estimation of the musical relevance of Newton's chromatic mode, and chromatic modes in general. As a possible direction of further investigation it is proposed to study the exo-mode of Newton's chromatic mode

New Hermite–Hadamard and Jensen inequalities for log-$$s$$-convex fuzzy-interval-valued functions in the second sense

In this paper, our aim is to consider the new class of log-convex fuzzy-interval-valued function known as log-s-convex fuzzy-interval-valued functions (log-s-convex fuzzy-IVFs). By this concept, we have introduced Hermite–Hadamard inequalities (HH-inequalities) by means of fuzzy order relation. This fuzzy order relation is defined level-wise through Kulisch–Miranker order relation defined on interval space. Moreover, some new Hermite–Hadamard–Fejér inequalities (HH–Fejér-inequalities) and Jensen’s inequalities via log-s-convex fuzzy-IVFs are also established and verified with the support of useful examples. Some special cases are also discussed which can be viewed as applications of fuzzy-interval HH-inequalities. The concepts and approaches of this paper may be the starting point for further research in this area.

New Hermite–Hadamard Inequalities in Fuzzy-Interval Fractional Calculus and Related Inequalities

It is a familiar fact that inequalities have become a very popular method using fractional integrals, and that this method has been the driving force behind many studies in recent years. Many forms of inequality have been studied, resulting in the introduction of new trend in inequality theory. The aim of this paper is to use a fuzzy order relation to introduce various types of inequalities. On the fuzzy interval space, this fuzzy order relation is defined level by level. With the help of this relation, firstly, we derive some discrete Jensen and Schur inequalities for convex fuzzy interval-valued functions (convex fuzzy-IVF), and then, we present Hermite–Hadamard inequalities (-inequalities) for convex fuzzy-IVF via fuzzy interval Riemann–Liouville fractional integrals. These outcomes are a generalization of a number of previously known results, and many new outcomes can be deduced as a result of appropriate parameter and real valued function selections. We hope that our fuzzy order relations results can be used to evaluate a number of mathematical problems related to real-world applications.

New Hermite–Hadamard-type inequalities for "Equation missing" No EquationSource Format="TEX", only image -convex fuzzy-interval-valued functions

In this paper, we introduce the non-convex interval-valued functions for fuzzy-interval-valued functions, which are called "Equation missing"-convex fuzzy-interval-valued functions, by means of fuzzy order relation. This fuzzy order relation is defined level-wise through Kulisch–Miranker order relation given on the interval space. By using the "Equation missing"-convexity concept, we present fuzzy-interval Hermite–Hadamard inequalities for fuzzy-interval-valued functions. Several exceptional cases are debated, which can be viewed as useful applications. Interesting examples that verify the applicability of the theory developed in this study are presented. The results of this paper can be considered as extensions of previously established results.

L-interval spaces and some of their properties

In this paper, notions of L-interval spaces and L-2-arity convex spaces are introduced. It is showed that there is a Galois’s connection between the category of L-convex spaces and the category of L-interval spaces. In particular, the category of L-2-arity convex spaces can be embedded in the category of L-interval spaces as a coreflective subcategory. Further, some properties of L-interval spaces are introduced including L-geometric (resp. L-Peano, L-Pasch and L-sand-glass) property. It is proved that an L-2-arity convex space is an L-JHC convex space iff its segment operator has L-Peano property. It is also proved that an L-JHC convex space with an L-idempotent segment operator has L-sand-glass property. Further, it is also proved that an L-idempotent interval space having L-Peano+L-Pasch property has L-geometric property and L-sand-glass property.

Some fuzzy-interval integral inequalities for harmonically convex fuzzy-interval-valued functions

<abstract> <p>It is well-known fact that fuzzy interval-valued functions (F-I-V-Fs) are generalizations of interval-valued functions (I-V-Fs), and inclusion relation and fuzzy order relation on interval space and fuzzy space are two different concepts. Therefore, by using fuzzy order relation (FOR), we derive inequalities of Hermite-Hadamard (<italic>H</italic>·<italic>H</italic>) and Hermite-Hadamard Fejér (<italic>H</italic>·<italic>H</italic> Fejér) like for harmonically convex fuzzy interval-valued functions by applying fuzzy Riemann integrals. Moreover, we establish the relation between fuzzy integral inequalities and fuzzy products of harmonically convex fuzzy interval-valued functions. The outcomes of this study are generalizations of many known results which can be viewed as an application of a defined new version of inequalities.</p> </abstract>

On one «Obscure» Link among Derivatives of Russ. zveno

The purpose of the article is to propose an etymology of Russ. dial. south взвéнье ‘wolf’s lair’. Existent hypotheses of the word’s etymology seem to be unconvincing in semantic or structural aspects. According to the author’s hypothesis the word is derived from звено. This idea is founded on the author’s reconstruction of initial semantics and genesis of Proto-Slav. *zveno. There are several hypotheses about the origin of Proto-Slav. *zveno, but its etymology remains obscure. In previous work devoted to semantic reconstruction the analysis of Slav. lexical material enabled the author to define repeating semantic elements in Slav. words as ‘part of the whole’ / ‘one of clamped parts’ / ‘space between clamps’. Then the denotatum of Proto-Slav. *zveno was defined as ‘interval, space, orifice with some filling’. So the hypothesis was suggested about the genesis of Proto-Slav. *zveno from Ind.-eur. *ĝhēη- ‘gape’. The defined denotatum of Proto-Slav. *zveno as interval, space allowed the author to suppose that Russ. dial. south взвéнье ‘wolf’s lair’ is derived from звено, with pref. в(ъ)- (instead of usually preferred въз-).