<abstract><title style='display:none'>Abstract</title>Let <italic>G</italic> = (<italic>V</italic> (<italic>G</italic>), <italic>E</italic>(<italic>G</italic>)) be a connected graph and <italic>x, y</italic> ∈ <italic>V</italic> (<italic>G</italic>), <italic>d</italic>(<italic>x, y</italic>) = min{ length of <italic>x − y</italic> path } and for <italic>e</italic> ∈ <italic>E</italic>(<italic>G</italic>), <italic>d</italic>(<italic>x, e</italic>) = min{<italic>d</italic>(<italic>x, a</italic>), <italic>d</italic>(<italic>x, b</italic>)}, where <italic>e</italic> = <italic>ab</italic>. A vertex <italic>x</italic> distinguishes two edges <italic>e</italic>1 and <italic>e</italic>2, if <italic>d</italic>(<italic>e</italic>1, <italic>x</italic>) ≠ <italic>d</italic>(<italic>e</italic>2, <italic>x</italic>). Let <italic>WE</italic> = {<italic>w</italic>1, <italic>w</italic>2, . . ., <italic>wk</italic>} be an ordered set in <italic>V</italic> (<italic>G</italic>) and let <italic>e</italic> ∈ <italic>E</italic>(<italic>G</italic>). The representation <italic>r</italic>(<italic>e | WE</italic>) of <italic>e</italic> with respect to <italic>WE</italic> is the <italic>k</italic>-tuple (<italic>d</italic>(<italic>e, w</italic>1), <italic>d</italic>(<italic>e, w</italic>2), . . ., <italic>d</italic>(<italic>e, wk</italic>)). If distinct edges of <italic>G</italic> have distinct representation with respect to <italic>WE</italic>, then <italic>WE</italic> is called an edge metric generator for <italic>G</italic>. An edge metric generator of minimum cardinality is an edge metric basis for <italic>G</italic>, and its cardinality is called edge metric dimension of <italic>G</italic>, denoted by edim(<italic>G</italic>). The circulant graph <italic>Cn</italic>(1, <italic>m</italic>) has vertex set {<italic>v</italic>1, <italic>v</italic>2, . . ., <italic>vn</italic>} and edge set {<italic>vivi</italic>+1 : 1 ≤ <italic>i</italic> ≤ <italic>n−</italic>1}∪{<italic>vnv</italic>1}∪{<italic>vivi+m </italic>: 1 ≤ <italic>i</italic> ≤ <italic>n−m</italic>}∪{<italic>vn−m+ivi</italic> : 1 ≤ <italic>i</italic> ≤ <italic>m</italic>}. In this paper, it is shown that the edge metric dimension of circulant graphs <italic>Cn</italic>(1, 2) and <italic>Cn</italic>(1, 3) is constant.</abstract>

Edge metric dimension of some classes of circulant graphs

<abstract><title style='display:none'>Abstract</title>Let (X; d) be a metric space endowed with a graph G such that the set V (G) of vertices of G coincides with X. We define the notion of G-Kannan maps and obtain a fixed point theorem for such mappings</abstract>

Fixed points of Kannan mappings in metric spaces endowed with a graph

<abstract><title style='display:none'>Abstract</title> We report on some advances made in the problem of singularities in general relativity.First is introduced the singular semi-Riemannian geometry for metrics which can change their signature (in particular be degenerate). The standard operations like covariant contraction, covariant derivative, and constructions like the Riemann curvature are usually prohibited by the fact that the metric is not invertible. The things become even worse at the points where the signature changes. We show that we can still do many of these operations, in a different framework which we propose. This allows the writing of an equivalent form of Einstein's equation, which works for degenerate metric too.Once we make the singularities manageable from mathematical viewpoint, we can extend analytically the black hole solutions and then choose from the maximal extensions globally hyperbolic regions. Then we find space-like foliations for these regions, with the implication that the initial data can be preserved in reasonable situations. We propose qualitative models of non-primordial and/or evaporating black holes.We supplement the material with a brief note reporting on progress made since this talk was given, which shows that we can analytically extend the Schwarzschild and Reissner-Nordström metrics at and beyond the singularities, and the singularities can be made degenerate and handled with the mathematical apparatus we developed. </abstract>

Spacetimes with singularities

<abstract><title style='display:none'>Abstract</title> In this article we characterize the ideal unimodular singularities in terms of their invariants. On the basis of this characterization we give an implementation of a classifier for ideal unimodular singularities in the computer algebra system SINGULAR.</abstract>

Department of Mathematics GC University, Faisalabad,

Department of Mathematical Sciences, Institute of Business Administration, Karachi,

Department of Mathematics, GC University, Lahore 54000,

A Classifier For Ideal Unimodular Singularities

<abstract><title style='display:none'>Abstract</title> In this paper, we first prove a path convergence theorem for a nonexpansive mapping in a reflexive and strictly convex Banach space which has a uniformly Gˆateaux differentiable norm and admits the duality mapping jφ, where φ is a gauge function on [0,∞). Using this result, strong convergence theorems for common fixed points of a countable family of nonexpansive mappings are established.</abstract>

School of Science, University of Phayao, Phayao 56000, Thailand

Department of Mathematics Education and the RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

Strong convergence theorems for a sequence of nonexpansive mappings with gauge functions

<abstract><title style='display:none'>Abstract</title> The goal of this note is to give an elementary characterization of the well-known quaternion group Q8 by using its subgroup lattice.</abstract>

A characterization of the quaternion group

<abstract>
<title style='display:none'>Abstract</title>
Let <italic>G</italic> and <italic>H</italic> be two graphs, each one of them being a path, a cycle or a star. In this paper, we determine the <italic>b</italic>-chromatic number of every subdivision-vertex neighbourhood corona <italic>G</italic> ⊡ <italic>H</italic> or <italic>G</italic> ⊡ <italic>K</italic>n, where <italic>K</italic>n is the complete graph of order <italic>n</italic>. It is also established for those graphs <italic>K</italic>n ⊡ <italic>G</italic> having <italic>m</italic>-degree not greater than <italic>n</italic> + 2. All the proofs are accompanied by illustrative examples.
</abstract>

Determining the <italic>b</italic>-chromatic number of subdivision-vertex neighbourhood coronas

<abstract>
<title style='display:none'>Abstract</title>
Ideals and filters are important notions with different meanings in the study of algebraic structures related to logical systems. In this paper we establish new connections between these concepts in residuated lattices.
</abstract>

Remarks on some connections between ideals and filters in residuated lattices

<abstract>
<title style='display:none'>Abstract</title>
Recent results in mathematical modeling predicting crack behavior under various load conditions in human bones as anisotropic elastic composite materials are presented in this survey. New and interesting challenges in theoretical models of fracture were proposed and had significant importance for fracture mechanics. Our goal is to present an overview of the use and limitations of existing relevant theories. The present study aims to introduce mathematical models to researchers unfamiliar with the concepts, to improve and provide new insights into bone fracture mechanics.
</abstract>

Recent advances of crack propagation in human bone

<abstract>
<title style='display:none'>Abstract</title>
We consider a generalization of the coupon collector problem with unequal probabilities, such that there are two additional coupons in the coupon set: one that speeds up the coupon collection process, and the one that slows it down. We derive some upper and lower bounds on the distribution function of the waiting time until a subcollection or a full collection of coupons is sampled.
</abstract>

Some bounds on the coupon collector problem with universal coupon

<abstract>
<title style='display:none'>Abstract</title>
This paper presents Leray–Schauder alternatives and a topological transversality (homotopy) theorem for compact upper semicontinuos maps having (strongly) convexly totally bounded ranges.
</abstract>

A Homotopy Theory for Maps Having Strongly Convexly Totally Bounded Ranges in Topological Vector Spaces

<abstract>
<title style='display:none'>Abstract</title>
In this study, we examine osculating-type curves with Frenet-type frame in Myller configuration for Euclidean 3-space <italic>E</italic>3. We present the necessary characterizations for a curve to be an osculating-type curve. Characterizations originating from the natural structure of Myller configuration are a generalization of osculating curves according to the Frenet frame. Also, we introduce some new results that are not valid for osculating curves. Then, we give an illustrative numerical example supported by a figure.
</abstract>

On generalized osculating-type curves in Myller configuration

<abstract><title style='display:none'>Abstract</title>In this paper, we introduce the notion of interpolative Rus-Reich-Ćirić type 𝒵- contractions in the setting of complete metric space. We also consider some immediate consequences of our main results.</abstract>

Interpolative Rus-Reich-Ćirić Type Contractions via Simulation Functions

<abstract><title style='display:none'>Abstract</title> We review the superiorization methodology, which can be thought of, in some cases, as lying between feasibility-seeking and constrained minimization. It is not quite trying to solve the full edged constrained minimization problem; rather, the task is to find a feasible point which is superior (with respect to an objective function value) to one returned by a feasibility-seeking only algorithm. We distinguish between two research directions in the superiorization methodology that nourish from the same general principle: Weak superiorization and strong superiorization and clarify their nature.</abstract>

Weak and Strong Superiorization: Between Feasibility-Seeking and Minimization

<abstract><title style='display:none'>Abstract</title>We consider a porous thermoelastic body, including voidage time derivative among the independent constitutive variables. For the initial boundary value problem of such materials, we analyze the temporal behaviour of the solutions. To this aim we use the Cesaro means for the components of energy and prove the asymptotic equipartition in mean of the kinetic and strain energies.</abstract>

Department of Mathematics and Computer Sciences, Transilvania University of Brasov, Bdul Iuliu Maniu, nr. 50, Brasov, Romania.

On Temporal Behaviour of Solutions in Thermoelasticity of Porous Micropolar Bodies

<abstract><title style='display:none'>Abstract</title>In this paper, the notion of closed cubic intuitionistic ideals, cubic intuitionistic <italic>p</italic>-ideals and cubic intuitionistic <italic>a</italic>-ideals in <italic>BCI</italic>-algebras are introduced, and several related properties are investigated. Relations between cubic intuitionistic subalgebras, closed cubic intuitionistic ideals, cubic intuitionistic <italic>q</italic>-ideals, cubic intuitionistic <italic>p</italic>-ideals and cubic intuitionistic <italic>a</italic>-ideals are discussed. Conditions for a cubic intuitionistic ideal to be a cubic intuitionistic <italic>p</italic>-ideal are provided. Characterizations of a cubic intuitionistic <italic>a</italic>-ideal are considered. The cubic intuitionistic extension property for a cubic intuitionistic <italic>a</italic>-ideal is established.</abstract>

Cubic Intuitionistic Structures Applied to Ideals of <italic>BCI</italic>-Algebras

<abstract><title style='display:none'>Abstract</title>The problem of reflection and transmission of plane waves at an imperfect boundary between two thermally conducting micropolar elastic solid half spaces with two temperature is investigated. Amplitude ratio of various reflected and transmitted waves are presented when a set of coupled longitudinal wave (LD-wave) and thermal wave (T-wave) and a set of coupled transverse wave and microrotation waves (CDI, CDII) is made incident. The expressions for reflection and transmission coefficients which are the ratios of the amplitudes of reflected and transmitted waves at different angles of incident wave are obtained. The corresponding expressions for the normal force stiffness, transverse force stiffness, transverse couple stiffness and perfect bonding has also been included. Stiffness and two temperature e ects on these amplitude ratios with angle of incidence have been depicted graphically. Some special and particular cases are also discussed.</abstract>

Department of Mechanical Engineering, NIT Kurukshetra, Kurukshetra 136119, India

Reflection and transmission of waves from imperfect boundary between two heat conducting micropolar thermoelastic solids

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