<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>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> 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> 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> In this article we introduce three centro-affine invariant functions in Minkowski spaces. Tzitzeica curves and Tzitzeica surfaces may be defined in this new context. Our main results allow us to understand the connections between Tzitzeica curves and surfaces in Minkowski spaces and the original Tzitzeica curves and surfaces from the Euclidian space</abstract>

Tzitzeica-Type centro-affine invariants in Minkowski spaces

<abstract>
<title style='display:none'>Abstract</title>
In this paper, we introduce the basic notions of the fractional summation, difference and <italic>q</italic>-difference with the quaternionic fractional order for the quaternion-valued functions and establish some of their basic properties. Based on this, the summation representations of solutions for the nonlinear quaternion-valued fractional difference equation and <italic>q</italic>-difference equation are obtained. In addition, several examples are provided to illustrate the feasibility of our obtained results in each section.
</abstract>

Quaternion Fractional Difference with Quaternionic Fractional Order and Applications to Fractional Difference Equation

<abstract>
<title style='display:none'>Abstract</title>
In this article, we study the existence of mild solutions of a non-instantaneous integrodi erential equations on unbounded domain via resolvent operators in Banach space. For our proofs, we employ the semigroups theory and Schauder’s fixed point theorem. Moreover, we show that solutions of our problem are attractive. Finally, an example is given to validate the theory part.
</abstract>

Existence and attractivity results on semi-infinite intervals for integrodifferential equations with non-instantaneous impulsions in Banach spaces

<abstract>
<title style='display:none'>Abstract</title>
Let <italic>n, m</italic> ∈ ℕ, <italic>n</italic> ≥ 3 and <italic>n</italic> ≥ <italic>m</italic> ≥ 2. In this paper, we study some properties of the unrestricted virtual singular braid of braid monoid <italic><bold>UVSB</bold>mn</italic> and group <bold><italic>UVSG</italic></bold><italic>mn</italic>. These properties emerged by studying the analogous properties of the unrestricted virtual singular pseudosymmetric braid of braid monoid and group. These mathematical objects are submonoids and subgroups of the virtual singular braid group <italic>VSGn</italic>. For many of these quotients, we have obtained reduced presentations.
</abstract>

On the unrestricted virtual singular braid

<abstract>
<title style='display:none'>Abstract</title>
Let Ω ⊂ ℝ<italic>N</italic>, <italic>N</italic> ≥ 2, be a bounded domain with smooth boundary ∂Ω. Consider the following generalized Robin-Steklov eigenvalue problem associated with the operator 𝒜<italic>u</italic> = <italic>−</italic> Δ<italic>pu −</italic> Δ<italic>qu</italic>
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_auom-2024-0003_eq_001.png"/>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mo>{</mml:mo> <mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>𝒜</mml:mi><mml:mi>u</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mrow><mml:mo>|</mml:mo> <mml:mi>u</mml:mi> <mml:mo>|</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>u</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mrow><mml:mo>|</mml:mo> <mml:mi>u</mml:mi> <mml:mo>|</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mrow><mml:mi>q</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mi>λ</mml:mi><mml:mi>α</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mrow><mml:mo>|</mml:mo> <mml:mi>u</mml:mi> <mml:mo>|</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mi> </mml:mi><mml:mi> </mml:mi><mml:mi> </mml:mi><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mfrac><mml:mrow><mml:mo>∂</mml:mo><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mi>A</mml:mi></mml:msub></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mrow><mml:mo>|</mml:mo> <mml:mi>u</mml:mi> <mml:mo>|</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>u</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mrow><mml:mo>|</mml:mo> <mml:mi>u</mml:mi> <mml:mo>|</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mrow><mml:mi>q</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mi>λ</mml:mi><mml:mi>β</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mrow><mml:mo>|</mml:mo> <mml:mi>u</mml:mi> <mml:mo>|</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mi> </mml:mi><mml:mi> </mml:mi><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mo>∂</mml:mo><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow> </mml:mrow></mml:mrow></mml:math>
<tex-math>\left\{ {\matrix{ {\mathcal{A}u + {\rho _1}\left( x \right){{\left| u \right|}^{p - 2}}u + {\rho _2}\left( x \right){{\left| u \right|}^{q - 2}}u = \lambda \alpha \left( x \right){{\left| u \right|}^{r - 2}}u,\,\,\,x \in \Omega ,} \cr {{{\partial u} \over {\partial {v_A}}} + {\gamma _1}\left( x \right){{\left| u \right|}^{p - 2}}u + {\gamma _2}\left( x \right){{\left| u \right|}^{q - 2}}u = \lambda \beta \left( x \right){{\left| u \right|}^{r - 2}}u,\,\,x \in \partial \Omega ,} \cr } } \right.</tex-math>
</alternatives>
</disp-formula>
where <italic>p, q, r</italic> ∈ (1, ∞), <italic>p &lt; q</italic>; α, <italic>ρi </italic>∈ <italic>L</italic>∞(Ω) and β, <italic>γi </italic>∈ <italic>L</italic>∞(∂Ω) are nonnegative functions satisfying ∫Ω <italic>α dx</italic> + ∫∂Ω β <italic>dσ &gt;</italic> 0 and ∫Ω ρ<italic>i dx</italic> + ∫∂Ω γ<italic>i dσ</italic> &gt; 0, <italic>i</italic> = 1, 2.
We show that, if either <italic>r &lt; p</italic> or <italic>r &gt; q</italic> with <italic>r &lt; q</italic>(<italic>N −</italic> 1)<italic>/</italic>(<italic>N − q</italic>) in case <italic>q &lt; N</italic>, then the eigenvalue set (spectrum) of the above problem is precisely (0, ∞). If <italic>r</italic> ∈ {<italic>p, q</italic>} then the corresponding spectrum is a smaller interval (<italic>d,</italic> ∞), <italic>d &gt;</italic> 0. On the other hand, if <italic>r</italic> ∈ (<italic>p, q</italic>) with <italic>r &lt; p</italic>(<italic>N −</italic> 1)<italic>/</italic>(<italic>N − p</italic>) in case <italic>p &lt; N</italic>, then we are able to identify an interval of eigenvalues [λ*, ∞), where λ* is a positive number depending on <italic>r</italic>.
Obviously, the spectrum of the above problem coincides with the spectra of the Neumann-like, Robin-like, and Steklov-like eigenvalue problems corresponding to the cases when some of the functions α, β, γ<italic>i </italic>vanish.
</abstract>

andreea.laura.burlacu@365.univ-ovidius.ro

On an eigenvalue problem associated with the (<italic>p, q</italic>) − Laplacian

<abstract>
<title style='display:none'>Abstract</title>
After introducing the definition of hypergroups by Marty, the study of hyperstructures and its connections with other fields has been of great importance. In this paper, we continue the investigation between hyperstructure theory and number theory. More precisely, we define an <italic>Hv</italic>-module of complex valued functions over the <italic>Hv</italic>-ring of arithmetics, classify its complete parts, the strongly regular relations on it and identify its fundamental module.
</abstract>

<italic>Hv</italic>-module of functions over <italic>Hv</italic>-ring of arithmetics and it’s fundamental module

<abstract>
<title style='display:none'>Abstract</title>
To extent the notion of the MP weak group inverse for square matrices, we introduce the concept of the weighted MP weak group inverse for rectangular matrices. A number of different representations and characterizations are derived for the weighted MP weak group inverse as well as limit and integral expressions. Applying the weighted MP weak group inverse, we solve some linear equations and give their solutions.
</abstract>

Weighted MP weak group inverse

<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 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> 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>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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