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We construct a two discrete parameter family of compact minimal surfaces embedded in the Berger sphere which may be considered as the analogue of the helicoidal Karcher-Scherk surfaces.

For an involution near ring 𝒩 and its ideal ℐ, the text introduces an involution ideal-based zero- divisor graph Γℐ*(𝒩) which is an undirected graph with vertex set { x ∈ 𝒩 - ℐ: x𝒩y ⊂ ℐ (or y𝒩x ⊂ ℑ ) for some y ∈ 𝒩-ℐ}, where two distinct vertices x and y are adjacent if and only if y𝒩x* ⊂ ℐ or x𝒩y* ⊂ ℐ. The paper provides characterizations of Γℐ*(𝒩) when it forms a complete graph or a star graph. It also explores the structure of Γℐ*(𝒩), investigates its properties like connectedness with diam(Γℐ*(𝒩)) ≤ 3 and analyzes the connection of Γℐ*(𝒩) with Γℐ*(𝒩/ℐ). Furthermore, the paper discusses the chromatic number and clique number of the graph. It also characterizes all right permutable *-near-rings 𝒩 for which the graph Γℐ*(𝒩) can be with a finite chromatic number.

n the present paper, the authors introduce a new subclass namely; RL(φ4L, ν) of Kamali-type starlike functions defined in the open unit disk D connected with four-leaf shaped domain. We investigate the bounds of some initial coefficients, Fekete-Szegö inequality and other results of logarithmic coefficients for the function belonging to above class. Relevant connections of the results derived in this paper with those of earlier works are indicated.

Strong convergence theorems that approximate common fixed points of two nonlinear mappings are presented. Our method is based on the Martinez-Yanes–Xu iteration, which extends Nakajo and Takahashi’s CQ method. In this paper, by exploiting the mean-valued iteration procedure, we further develop Nakajo and Takahashi’s CQ method and Takahashi, Takeuchi, and Kubota’s shrinking projection method. The approach of this paper does not require that the two mappings be continuous or commutative. The types of mappings considered in this paper include nonexpansive mappings and other well-known classes of mappings as special cases.

In this paper we continue our investigation concerning the concept of a liken. This notion has been defined as a sequence of non-negative real numbers, tending to infinity and closed with respect to addition in R. The most important examples of likens are clearly the set of natural numbers N with addition and the set of positive natural numbers N* with multiplication, represented by the sequence (ln(n+1))∞ n=0. The set of all likens can be parameterized by the points of some infinite dimensional, complete metric space. In this space of likens we consider elements up to isomorphism and define properties of likens as such that are isomorphism invariant. The main result of this paper is a theorem characterizing the liken N* of natural numbers with multiplication in the space of all likens.

In 1985 Crapo introduced in [3] a new mathematical object that he called geometry of circuits. Four years later, in 1989, Manin and Schechtman defined in [13] the same object and called it discriminantal arrangement, the name by which it is known now a days. Those discriminantal arrangements $\mathcal{B}(n,k,\mathcal{A}^0)$ are builded from an arrangement $\mathcal{A}^0$ of $n$ hyperplanes in general position in a $k$-dimensional space and their combinatorics depends on the arrangement $\mathcal{A}^0$. On this basis, in 1997 Bayer and Brandt (see [2]) distinguished two different type of arrangements $\mathcal{A}^0$ calling very generic the ones for which the intersection lattice of $\mathcal{B}(n,k,\mathcal{A}^0)$ has maximum cardinality and non-very generic the others. Results on the combinatorics of $\mathcal{B}(n,k,\mathcal{A}^0)$ in the very generic case already appear in Crapo [3] and in 1997 in Athanasiadis [1] while the first known result on non-very generic case is due to Libgober and the first author in 2018. In their paper [12] they provided a necessary and sufficient condition on $\mathcal{A}^0$ for which the cardinality of rank 2 intersections in $\mathcal{B}(n,k,\mathcal{A}^0)$ is not maximal anymore. In this paper we further develop their result providing a sufficient condition on $\mathcal{A}^0$ for which the cardinality of rank r, $r \ge 2$, intersections in $\mathcal{B}(n,k,\mathcal{A}^0)$ decreases.

The classic method of solving the cubic and the quartic equations using Tschirnhaus transformation yields true as well as false solutions. Recently some papers on this topic are published, in which methods are given to get only the true solutions of cubic and quartic equations. However these methods have some limitations. In this paper the author presents a method of solving cubic and quartic equations using Tschirnhaus transformation, which yields only the true solutions. The proposed method is much simpler than the methods published earlier.

The purpose of the present paper is to investigate the Sălăgean - type probability distribution of order α. In this paper, we obtain moments, factorial moments and moment generating function for this distribution. Finally, we obtain a entropy of this distribution. Our results improve and generalize the results of Porwal [Starlike and convex type probability distribution, Afr. Mat. 30 (2019)].

Anantharaman and Le Masson proved that any family of eigenbases of the adjacency operators of a family of graphs is quantum ergodic (a form of delocalization) assuming the graphs satisfy conditions of expansion and high girth. In this paper, we show that neither of these two conditions is sufficient by itself to necessitate quantum ergodicity. We also show that having conditions of expansion and a specific relaxation of the high girth constraint present in later papers on quantum ergodicity is not sufficient. We do so by proving new properties of the Cartesian product of two graphs where one is infinite.

The paper summarizes the main core of the last results that we obtained in [8, 4, 17] on the regularity of the value function for a Bolza problem of a one-dimensional, vectorial problem of the calculus of variations. We are concerned with a nonautonomous Lagrangian that is possibly highly discontinuous in the state and velocity variables, nonconvex in the velocity variable and non coercive. The main results are achieved under the assumption that the Lagrangian is convex on the one-dimensional lines of the velocity variable and satisfies a local Lipschitz continuity condition w.r.t. the time variable, known in the literature as Property (S), and strictly related to the validity of the Erdmann–Du-Bois Reymond equation.Under our assumptions, there exists a minimizing sequence of Lipschitz functions. A first consequence is that we can exclude the presence of the Lavrentiev phenomenon. Moreover, under a further mild growth assumption satisfied by the minimal length functional, fully described in the paper, the above sequence may be taken with the same Lipschitz rank, even when the initial datum and initial value vary on a compact set. The Lipschitz regularity of the value function follows.

The last three decades two distinct trends have emerged within Early Childhood Education, namely Early Childhood Science Education and Early Childhood Special Education. The current paper tries to move along these areas and illuminate the common ground that lie between them. In particular, drawing from the formulated empiricist, Piagetian and socio-cognitive trends within Early Childhood Science Education, it tries to explore to what extend they could expand in special needs context. In addition, it proposes diverse, specific strategies that emerge from the above-mentioned trends and could be implemented in teaching science within special needs settings. By so doing, the present paper aspires to set the frame of a newly born and well-promising trend, that of Early Childhood Special Science Education.

In this paper we present a short overview of results that provide relationships among scalar, matrix-valued and certain operator-valued positive definite kernels. We refine and extend some of them in order that they may be applied for strict positive definiteness as well. This is a topic not well explored in the literature but that has potential usefulness in the characterization of several classes of positive definite and strictly positive definite kernels. This is ratified in the paper with the inclusion of a number of applications and examples.

The purpose of this paper is to establish necessary and sufficient optimality conditions and some duality results for weakly efficient solutions of a constrained bilevel multiobjective fractional programming problem (P) with an extremal-value function. Using parametric approach, the problem (P) is first equivalently transformed into a parametric problem (P µ) with µ ∈ Rp, for which we construct then a dual problem. This is achieved in terms of conjugate duality theory. Under appropriate assumptions, the weak and strong duality results for (Pµ) are presented. These results permit us to give dual characterizations for the weakly efficient solutions of the problem (P).

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The bicategory of Landau-Ginzburg models denoted by ℒ𝒢K possesses adjoints and this helps in explaining a certain duality that exists in the setting of Landau-Ginzburg models in terms of some specified relations. The construction of ℒ𝒢K is reminiscent of, but more complex than, the construction of the bicategory of associative algebras and bimodules. In this paper, we review this complex but very inspiring construction in order to expose it more to pure mathematicians. In particular, we spend some time explaining the intricate construction of unit morphisms in this bicategory from a new vantage point. Besides, we briefly discuss how this bicategory could be constructed in more than one way using the variants of the Yoshino tensor product. Furthermore, without resorting to Atiyah classes, we prove that the left and right unitors in this bicategory have direct right inverses but do not have direct left inverses.

The objective of this paper is to generalize and improve some results in fixed point theorems in both complete metric space and Menger space. These results are generalizations of the analogous ones recently proved by Khojasteh [5], Demma [1], Yildirim [13], where we establish a dynamic information about their other fixed points if there exist i.e the distance between two fixed point in case of metric space and their equivalent in probabilistic metric space. Some illustrative examples are furnished, which demonstrate the validity of the hypotheses. As an application to our main result, we derive a uniqueness fixed point theorem for a self-mapping under strong conditions.

Our aim in this paper is to show that the modulus of smoothness and the $K$-functionals constructed from the Sobolev-type space corresponding to the Dunkl operator are equivalent on the interval $(-1,1)$.

Let $R$ be a commutative noetherian local ring which is singular and has an isolated singularity. Let $\mathsf {D_{sg}}(R)$ be the singularity category of $R$ in the sense of Buchweitz and Orlov. In this paper, we find real numbers $t$ such that the complexity $\delta _t(G,X)$ in the sense of Dimitrov, Haiden, Katzarkov and Kontsevich vanishes for any split generator $G$ of $\mathsf {D_{sg}}(R)$ and any object $X$ of $\mathsf {D_{sg}}(R)$. In particular, the entropy $\mathrm{h}_t(F)$ of an exact endofunctor $F$ of $\mathsf {D_{sg}}(R)$ is not defined for such numbers $t$.

Understanding the behavior of a black-box model with probabilistic inputs can be based on the decomposition of a parameter of interest (e.g., its variance) into contributions attributed to each coalition of inputs (i.e., subsets of inputs). In this paper, we produce conditions for obtaining unambiguous and interpretable decompositions of very general parameters of interest. This allows recovering known decompositions, holding under weaker assumptions than the literature states.

In this paper, we study the relative homological dimension based on the class of projectively coresolved Gorenstein flat modules (PGF-modules), that were introduced by Saroch and Stovicek in [26]. The resulting PGF-dimension of modules has several properties in common with the Gorenstein projective dimension, the relative homological theory based on the class of Gorenstein projective modules. In particular, there is a hereditary Hovey triple in the category of modules of finite PGF-dimension, whose associated homotopy category is triangulated equivalent to the stable category of PGF-modules. Studying the finiteness of the PGF global dimension reveals a connection between classical homological invariants of left and right modules over the ring, that leads to generalizations of certain results by Jensen [24], Gedrich and Gruenberg [17] that were originally proved in the realm of commutative Noetherian rings.

The optimal transport map between the standard Gaussian measure and an $\alpha $-strongly log-concave probability measure is $\alpha ^{-1/2}$-Lipschitz, as first observed in a celebrated theorem of Caffarelli. In this paper, we apply two classical covariance inequalities (the Brascamp–Lieb and Cramér–Rao inequalities) to prove a sharp bound on the Lipschitz constant of the map that arises from entropically regularized optimal transport. In the limit as the regularization tends to zero, we obtain an elegant and short proof of Caffarelli’s original result. We also extend Caffarelli’s theorem to the setting in which the Hessians of the log-densities of the measures are bounded by arbitrary positive definite commuting matrices.

This paper studies the unique continuation problem for the heat equation. We prove a so-called conditional stability estimate for the solution. We are interested in local estimates that are Hölder stable with the weakest possible norms of data on the right-hand side. Such an estimate is useful for the convergence analysis of computational methods dealing with data assimilation. We focus on the case of a known solution at initial time and in some subdomain but that is unknown on the boundary. To the best of our knowledge, this situation has not yet been studied in the literature.

The last decade has seen an abundance of congruences for $b_\ell (n)$, the number of $\ell $-regular partitions of $n$. Notably absent are congruences modulo $4$ for $b_3(n)$. In this paper, we introduce Ramanujan type congruences modulo $4$ for $b_3(2n)$ involving some primes $p$ congruent to $ 11, 13, 17, 19, 23$ modulo $24$.

A conjecture by the second author, proven by Bonnafé–Rouquier, says that the multiplicity matrix for baby Verma modules over the restricted rational Cherednik algebra has rank one over $\mathbb{Q}$ when restricted to each block of the algebra.In this paper, we show that if $H$ is a prime algebra that is a free Frobenius extension over a regular central subalgebra $R$, and the centre of $H$ is normal Gorenstein, then each central quotient $A$ of $H$ by a maximal ideal $\mathfrak{m}$ of $R$ satisfies the rank-one property with respect to the Cartan matrix of $A$. Examples where the result is applicable include graded Hecke algebras, extended affine Hecke algebras, quantized enveloping algebras at roots of unity, non-commutative crepant resolutions of Gorenstein domains and 3 and 4 dimensional PI Sklyanin algebras.In particular, since the multiplicity matrix for restricted rational Cherednik algebras has the rank-one property if and only if its Cartan matrix does, our result provides a different proof of the original conjecture.

In this paper, we consider the class of normalized analytic functions of the form f(z)=z+∑n=2∞ anzn. Following this functions, we define the functions whose coefficients are probabilities of the geometric distribution series and other special modes of this series. Also, we consider different special classes of f(z). In the following we consider some lemmas that make connection between defined special classes with the function f(z). Follower of this topic we will consider the theorems that make connection between defined classes with the functions whose coefficients are probabilities of geometric distribution series. Also we define Alexander-type integral operator and find the necessary and sufficient conditions for being this operator to defined general classes.

In this paper, we will present the author's interpretation and embellishment of five lectures on cluster theory given by Kiyoshi Igusa during the Spring semester of 2022 at Brandeis University. They are meant to be used as an introduction to cluster theory from a representation-theoretic point of view.

Student motivation is one of the affective components that plays a key role in learning science. Research shows that in developed countries there is a trend of declining students’ interest in science and technology. In this paper, the relationships between intrinsic motivation, self-concept, and the achievement of the fourth-grade elementary school students in mathematics and sciences were investigated. The authors also present the results of the secondary data analyses obtained in Serbia by means of student questionnaires and knowledge tests in the last two cycles of the TIMSS research in 2015 and 2019. The questionnaire contains two scales in which motivational variables are operationalized: students’ attitudes towards mathematics/sciences and mathematical/ scientific self-confidence. Students’ attitude is an indicator of an intrinsic motivation, and self-confidence is an indicator of self-concept. The results show that students express a high motivation for learning mathematics and sciences and a high level of self-concept. A higher level of self-concept is accompanied by a higher level of motivation for learning mathematics and sciences. Individual characteristics of students have a greater influence on achievement than the influence of schools and teachers. Students who express a more positive attitude towards mathematics and science and a higher level of self-concept in these areas also have a higher level of achievement. The influence of mathematical self-concept on achievement is particularly significant. The obtained results are in accordance with the findings of the earlier research that indicate complex and controversial relationships between intrinsic motivation, self-concept, and mathematical achievement, as well as that other constructs, including self-concept, can mediate the links between motivation and achievement. The authors point out the possibilities of encouraging students’ motivation for learning. If students are trained in the teaching process to organize their own activities, this will contribute to their experience of autonomy and the development of confidence in their own competence, which positively impacts their motivation for learning.

Let $\mathrm{F}$ be a non-archimedean locally compact field and let $\mathrm{G}_{n}$ be the group $\mathrm{PGL}_{n}(\mathrm{F})$. In this paper we construct a tower $(\tilde{\mathrm{X}}_{k})_{k\geqslant 0}$ of graphs fibred over the one-skeleton of the Bruhat–Tits building of $\mathrm{G}_{n}$. We prove that a non-spherical and irreducible generic complex representation of $\mathrm{G}_{n}$ can be realized as a quotient of the compactly supported cohomology of the graph $\tilde{\mathrm{X}}_{k}$ for $k$ large enough. Moreover, when the representation is cuspidal then it has a unique realization in a such model.

In this paper, we study the alternating Euler $T$-sums and $\tilde{S}$-sums, which are infinite series involving (alternating) odd harmonic numbers, and have similar forms and close relations to the Dirichlet beta functions. By using the method of residue computations, we establish the explicit formulas for the (alternating) linear and quadratic Euler $T$-sums and $\tilde{S}$-sums, from which, the parity theorems of Hoffman’s double and triple $t$-values and Kaneko–Tsumura’s double and triple $T$-values are further obtained. As supplements, we also show that the linear $T$-sums and $\tilde{S}$-sums are expressible in terms of colored multiple zeta values. Some interesting consequences and illustrative examples are presented.

In this paper, we present certain results concerning the distribution of zeros of polynomials of a quaternionic variable and with quaternionic coefficients. We obtain ring shaped regions of Eneström–Kakeya type for the zeros of these polynomials and also extend some classical results from the complex to quaternionic setting.

Currently, researchers have indicated that inorganic minerals in reservoirs, such as clay minerals, carbonates and quartz, can catalyze the evolution of organic matter into oil and gas. Therefore it is reasonable to believe that the minerals in reservoirs may act as a catalyst support with the metal-containing catalyst added from outside during the thermal recovery of heavy oil. This paper studied the aquathermolysis of heavy oil catalyzed by minerals and transition metal. The reaction conditions of two heavy oil samples were investigated. The results show that the optimal reaction conditions of heavy oil from Xinjiang Baikouquan Oilfield (XBO) are the reaction temperature of 250 °C and the reaction time of 6 h; for the crude oil from Xinjiang Tahe Oilfield (XTO), the optimal reaction conditions are determined to be the reaction temperature of 250 °C and the reaction time of 12 h, the water–oil ratio of the two oils is 0.3. Under optimal conditions, viscosity and pour point of heavy oil are significantly reduced. Differential scanning calorimetry (DSC), GC-MS analysis, thermogravimetric analysis (TGA), and elemental analysis were used to study the properties of the two heavy oil samples before and after reaction to explore the mechanism of the catalyzed aquathermolysis of heavy oil. This work will benefit the related heavy oil recovery work in this field.

Let R be a commutative ring with identity, M an R-module and S ⊆ R a multiplicative set. Then M is called S-finite if there exist an s ∈ S and a finitely generated submodule N of M such that sM ⊆ N. Also, M is called S-Noetherian if each submodule of M is S-finite. A ring R is called S-Noetherian if it is S-Noetherian as an R-module. This paper surveys the most recent developments in describing the structural properties of S-Noetherian rings, S-Noetherian modules and their generalizations. Some interesting constructed examples of S-Noetherian rings and modules are also presented.

In this paper, the nonlinear shallow water problem is studied analytically and numerically. This problem has been studied by different authors in the linear case; here we consider a nonlinear one with non constant coefficients. We first start by a mathematical analysis, where the well posedness is proved via the use of traveling wave solutions. The existence and uniqueness of the solution is demonstrated by the means of Schaulder's and Banach's fixed point theorems. Then, we explore the proposed model and investigate the stability conditions and the efficiency numerically.

In this paper we introduce the concept of positive Cohen p-nuclear polynomials between Banach lattice spaces. We give an analogue to Pietsch domination theorem and we study some properties concerning this notion.

In this paper, we prove some Hardy type and Hardy-Steklov type integral inequalities for two negative summation parameters and we deduce some well-known results with sharp constants.

In this paper, we focus on some class of Stefan type problems. We prove the existence and uniqueness of renormalized solution in anisotropic Sobolev spaces with data belonging to L1 -data, based on the properties of the renormalized solutions and the generalized monotonicity method in the functional spaces.

In this paper, we consider a general single species model in a heterogeneous environment of $n$ patches ($n\ge 2$), where each patch follows a generalized logistic law. First, we prove the global stability of the model. Second, in the case of perfect mixing, i.e. when the migration rate tends to infinity, the total population follows a generalized logistic law with a carrying capacity which in general is different from the sum of the $n$ carrying capacities. Next, we give some properties of the total equilibrium population and we compute its derivative at no dispersal. In some particular cases, we determine the conditions under which fragmentation and migration can lead to a total equilibrium population which might be greater or smaller than the sum of the $n $ carrying capacities. Finally, we study an example of two-patch model where the first patch follows a logistic law and the second a Richard’s law, we give a complete classification of the model parameter space as to whether dispersal is beneficial or detrimental to the sum of two carrying capacities.

Student knowledge assessment is a key element of the pedagogical process, as it provides students, parents, and educators with important feedback information on students’ knowledge and skills. Assessment of students’ mathematical knowledge is complex, as several factors are normally included in students’ final grade, and simply calculating the average of students’ achievements may not provide a complete picture of their knowledge. Therefore, in this paper, we aimed to investigate the possibility of using fuzzy logic to assess students’ knowledge and competencies by considering (1) students’ overall academic performance, (2) the quality of students’ project work on a topic from the history of mathematics and (3) the regularity of handing in homework. The study was conducted considering 22 Italian high school students. The results show that students’ academic performance is similar to student grades obtained with fuzzy logic.

This paper aims to describe and explain the impact of certain teaching practices on collegen students’ conception changes. Teachers' practices during a lesson on atomic models were studied, as well as students' conceptions, based on diagrams they drew and commented, before and after the lesson. Aware of the difficulty of changing these students’ conceptions, teachers deliberately use certain forms of content representations as well teaching activities designed to change them. The students' conceptions generally evolve towards a more advanced level of formulation corresponding to the probabilistic model, even if several errors still remain.

This paper presents an approach to characterising the professional identity of technical vocational teachers based on psychosociological indicators. The indicators are based on the scientific literature and refer to the representation of colleagues and the profession to oneself and the representation of oneself as a person. A questionnaire survey was submitted to two groups of teachers with several years of service. The first group was made up of teachers with a professional degree. The second group consisted of contract teachers who teach without a professional diploma. This study shows the existence of several identity profiles of technical vocational training (TVET) teachers in Senegal.

In this paper we consider the following moment problem: find a positive Borel measure μ on ℂ subject to conditions ∫ zn dμ = sn, n∈ℤ+, where sn are prescribed complex numbers (moments). This moment problem may be viewed (informally) as an extension of the Stieltjes and Hamburger moment problems to the complex plane. A criterion for the moment problem for the existence of a compactly supported solution is given. In particular, such moment problems appear naturally in the domain of complex Jacobi matrices. For every bounded complex Jacobi matrix its associated functional S has the following integral representation: S(p) = ∫ℂ p(z) dμ, with a positive Borel measure μ in the complex plane. An interrelation of the associated to the complex Jacobi matrix operator A0, acting in l2 on finitely supported vectors, and the multiplication by z operator in L2μ is discussed.

In this paper, we study semilinear conformable fractional evolution equations with finite delay subjected to nonlocal initial conditions in an arbitrary Banach space. We prove the existence of mild solutions under compactness type conditions on the nonlinear forcing term. Our result improves and complements several earlier related works. We apply our result to study a functional conformable partial differential equation of transport type.

In the present paper, we determine necessary and sufficient conditions for the subclass T(α, b) of analytic functions associated with Pascal distribution. Further, we consider properties of a special function related to Pascal distribution series. Several corollaries and consequences of the main results are also considered.

Let X be a projective, equidimensional, singular scheme over an algebraically closed field. Then the existence of a geometric smoothing (i.e. a family of deformations of X over a smooth base curve whose generic fibre is smooth) implies the existence of a formal smoothing as defined by Tziolas. In this paper we address the reverse question giving sufficient conditions on X that guarantee the converse, i.e. formal smoothability implies geometric smoothability. This is useful in light of Tziolas’ results giving sufficient criteria for the existence of formal smoothings.

In this paper, a new family of rotationally symmetric planar graphs is described based on an edge coalescence of planar chorded cycles. Their local fractional metric dimension is established for those ones arisen from chorded cycles of order up to six. Their asymptotic behaviour enables us to ensure the existence of new families of rotationally symmetric planar graphs with either constant or bounded local fractional dimension.

In this paper we develop two approaches for studying a large family of generalized Bernoulli–Euler polynomials. For the determinental approach, using Little Fermat’s Theorem, we establish a congruence identity and we give an explicit formulas of the generalized Bernoulli–Euler polynomials in terms of the Stirling numbers. The linear recursive approach allows us to formulate some properties of the generalized Bernoulli–Euler numbers and the generalized Bernoulli–Euler polynomials. Moreover, combinatorial formulas for these polynomials are provided.

In this paper, we introduce and study a set relative to singularities of plurisubharmonic functions. We prove that this set is countable under the condition $h>0$ on $\mathbb{B}\setminus \lbrace 0\rbrace .$

The present paper deals with lines contained in a smooth complex cubic threefold. It is well-known that the set of lines of the second type on a cubic threefold is a curve on its Fano surface. Here we give a description of the singularities of this curve.

In this paper we establish a result in the fixed point theory of multivalued mappings. This is done by combining three prevalent trends in fixed point theory. Some consequences of the main theorem are discussed. We also provide an illustrative example. The results are derived on metric spaces with a relation.

The content of this paper is a generalization of a theorem by Joseph Rabinoff: if $\mathscr{P}$ is a finite family of pointed and rational polyhedra in $N_\mathbb{R}$ such that there exists a fan in $N_\mathbb{R}$ that contains all the recession cones of the polyhedra of $\mathscr{P}$, if $k$ is a complete non-archimedean field, if $S$ is a connected and regular $k$-analytic space (in the sense of Berkovich) and $Y$ is a closed $k$-analytic subset of $U_{\mathscr{P}} \times _k S$ which is relative complete intersection and contained in the relative interior of $U_{\mathscr{P}} \times _k S$ over $S$, then the quasifiniteness of $\pi : Y \rightarrow S$ implies its flatness and finiteness; moreover, all the finite fibers of $\pi $ have the same length. This namely gives a analytic justification to the concept of stable intersection used in the theory of tropical intersection.