In this paper, we introduce new definitions of extended fuzzy T1 spaces and establish relations
between them and their counterparts. We show that these concepts have projective, productive, and hereditary
characteristics. We also demonstrate that generalized bijective fuzzy continuous and generalized fuzzy open
mappings preserve these spaces. Furthermore, these ideas are examined in the framework of initial and final
extended fuzzy topological spaces.

The development of second-generation (2G) bioethanol from lignocellulosic sources, such as sugarcane bagasse, is
very important as a viable alternative to conventional fossil fuels. However, the high cost associated with enzymatic
hydrolysis, which breaks down cellulose into fermentable sugars, poses a key challenge. This study focused on
enhancing cellulase enzyme production by a novel, locally isolated strain, Trichoderma harzianum PP400831,
using statistical optimization BBD-RSM to improve enzyme activity. Optimization efforts resulted in maximal
endoglucanase and exoglucanase activities of 4.01 IU/mL and 2.64 IU/mL, respectively after 9 days at 2% cellulose
mixture concentration and 0.15% tween 80. After saccharifcation of pretreated (SCB) by the crude enzymes and
fermentation of produced reduced sugar by S. cerevisiae MN901244 yielded an ethanol concentration of 25.63
g/L. This work represents a signifcant step toward developing a cost-effective, sustainable, and high-performing
cellulase production process for second-generation bioethanol. 
 

Line congruences are crucial in classical geometry, particularly in relating one surface to another through families of lines. These correspondences
are most valuable when they preserve key geometric features of the original surface. A line congruence, understood as a two-parameter
family of lines, can itself be viewed as a surface within the space of lines. This paper focuses on timelike line congruences, using the Study
map to explore their geometry within Minkowski 3-space. By interpreting a timelike line congruence as a region on the hyperbolic dual unit
sphere, we connect surface theory with the geometry of these congruences. We introduce the first and second fundamental forms to establish
conditions for when a timelike surface is developable and to study its differential properties. Applying Blaschke’s moving frame technique, we
derive curvature formulas and provide Minkowski analogs of classical results for ruled surfaces within the congruence. Specifically, we extend
known Euclidean results, including a Minkowski version of Plücker’s conoid. We also derive Dupin’s indicatrix for timelike line congruences,
offering a classification based on curvature invariants. In addition, we construct the Liouville formula within this framework and discuss its
geometric implications for closed timelike ruled surfaces contained in a timelike line congruence. To highlight the practical outcomes of our
approach, we provide several illustrative models.

This study explores the geometry of timelike ruled surfaces and their associated Blaschke frames in Minkowski 3-space. It establishes a
mapping from spacelike differentiable curves to timelike ruled surfaces and derives the corresponding differential equations governing the
Blaschke frame, which encapsulates key geometric vectors of the surface. Central concepts, such as the striction curve and the Disteli-axis,
are analyzed, highlighting their roles in the surface’s motion and curvature. The research further investigates the conditions for rotational
and translational motions and classifies ruled surfaces based on specific curvature and torsion constraints. Overall, this study offers a
comprehensive framework for understanding the geometry and kinematics of timelike ruled surfaces.

This study investigates the geometric properties of slant timelike-ruled surfaces and their Bertrand offsets in Minkowski space. By deriving
their parametric equations, we examine the structural characteristics of these surfaces and classify their offset relationships. Through the use
of geodesic curvatures, we establish conditions for parallel Bertrand offsets and analyze their compatibility with the Blaschke frame. Explicit
representations of the slant timelike-ruled surface and its Bertrand offset are formulated, with specific parameter values chosen to explore their
geometric behavior. The influence of these parameters on surface geometry is demonstrated through graphical models. These results advance
the understanding of ruled surface theory in Lorentzian geometry and offer valuable insights into applications in mathematical physics and
differential geometry.

In thisstudy,weintroducetheconceptofstationary-angletimelike-ruledsurfacesandexaminetheirgeometricproperties,
particularly inrelationtotheirBertrandoffsets.Atimelike-ruledsurfaceisgeneratedbythemotionofastraightrulingalonga
striction curve,anditsstructureisanalyzedusingtheBlaschkeandDarbouxframes.Wederivekeygeometricinvariants,including
spherical curvature,geodesiccurvature,normalcurvature,andgeodesictorsion.Additionally,weestablishtheconditionsunder
which thestrictioncurveofatimelike-ruledsurfacebehavesasageodesic,anasymptoticcurve,oracurvatureline.Specialcases,
such astimelike-tangentialdevelopablesandtimelike-cones,arealsodiscussed.Usingcurvature-axisanalysis,wedevelopahigher-
order contactframeworktobetterunderstandthebehaviorofthesesurfaces.Finally,weinvestigatetheBertrandoffsetsof
stationary-angle timelike-ruledsurfaces,provingthattheypreserveastationaryanglebetweentheirrulingsanddetermining
the necessaryconditionsfortheirexistence.ThisworkenhancestheunderstandingofdifferentialgeometryinLorentzianspaces
and providesnewinsightsintoruledsurfacesinMinkowskispace.

This study investigates fixed-axis spacelike ruled surfaces and their evolute offset counterparts
within E31
(Minkowski 3-space). The analysis utilizes the Blaschke frame associated
with the striction curves of these surfaces. Spacelike ruled surfaces play a crucial
role in various fields of both classical and modern physics. The research begins by introducing
the fundamental concepts of fixed-axis spacelike ruled surfaces and defining a
height function that establishes the necessary criteria for a ruled surface to be classified
as a fixed-axis spacelike ruled surface. Subsequently, the study derives parameterization
for both the fixed-axis spacelike ruled surfaces and their evolute offsets. Finally, several
surface models are extended and visually represented through graphical illustrations.

In this paper, we propose a method for constructing families of spacelike
surfaces in Minkowski 3-space 𝔼31
that share Bertrand curves as asymptotic
curves. By using marching-scale functions, we derive the necessary conditions,
provide flexible formulations, and establish a framework for constructing mutual
spacelike Bertrand curves. Examples show how different functions generate
surfaces interpolating the common asymptotic curves, offering new insights for
geometric modeling and ruled surface theory.

The developable surface (DS) is a curved surface that can be spread out on a plane without stretching or tearing, which is
widely operated in much fields of engineering and industrialization. This research displays a new approach of producing developable
surfaces in E3(Euclidean 3-space). At first, we start a modified frame over a curve, named as the quasi-frame. We then initiate an
exemplification of a DS and call it a quasi-normal DS. At the essence of this work, we examine the existence and uniqueness of such
DS, then consider its categorizations via singularity theory and unfolding theory (UT). Finally, two paradigms related to our approach
are presented for the purpose of clarity.