Algebraic structures on topological spaces can’t distinguish all the non-homeomorphic spaces. Recently, by equipping some of these algebraic structures with topology, one can separate the non-homeomorphic spaces with the same algebraic structures. Particularly, the fundamental group is equipped by the various types of topologies which some of them make it a topological group and some of them don't‎. Virk‎ ‎and‎ ‎Zastro‎w has‎ ‎studied‎ ‎the‎ ‎generalizations‎ ‎of‎ ‎the‎ ‎existing‎‎‎ ‎topologies‎ ‎on‎ ‎the‎ ‎fundamental‎‎group‎ ‎to‎ ‎the‎ ‎universal‎ ‎path‎ ‎space‎ with a proper comparison. If we remove the condition α(0) = x0  from the definition of the universal path space,  we obtain an object that have been discussed in mathematical literatures under the name  “ fundamental groupoid”. Indeed, fundamental groupoid denoted by πX , is the category of homotopy classes of paths in X  as the morphisms and has the set X  as the objects set. For any x , y  ∈X , the set πXx,y  is the set of homotopy classes of paths in X  from x  to y . We can consider the object group at x , πXx , as the well-known fundamental group π1X,x.  ‎ ‎ In order to answer the question how these topologies can be generalized on the fundamental groupoids, the authors has introduced the Lasso topology on the fundamental groupoid of a locally path connected space in which makes it a topological groupoid. A topological groupoid is a groupoid G  together with topologies on G  and G0  such that the structure maps are continuous‎.  R‎. ‎Brown and G‎. ‎Danesh-Naruie were the first and only ones to take this step‎. ‎They have defined a topology on a quotient of the fundamental groupoid such that it became a topological groupoid when the given space X  is locally path connected and semilocally simply connected. Here, we introduce whisker topology on the fundamental groupoid of a locally path connected space X  in which it’s basis is known and by some assumptions, we can consider it as a generalization of the whisker topology on the fundamental group. Material and methods ‎‎ For a given topological space (X,τ) , let [α]∈πX(x,y) ‎, where x,y∈X . ‎If V , ‎ W  are open neighborhoods of x ‎, y, respectively‎, one can define Nα,V,W:=β∈πX β≃γ*α*λ, γI⊆V, λI⊆W} , where γ1=α0=x  and α1=λ0=y. Theorem: ‎The family‎ ‎Nα,V,W;α∈πXx,y,x∈V∈τ,y∈W∈τ ‎forms a basis for a topology on fundamental groupoid‎.  ‎The topology that is generated by this basis‎, ‎is called Whisker topology. Conclusion Based on the results that we presented in this paper: If X  is a small loop transfer space; The multiplication map m: πwhX × πwhX → πwhX is continuous. The fundamental groupoid with the Whisker topology is a topological groupoid. The inherited topology from fundamental groupoid πwhX  on the object group πwhX(x)  equals to the Whisker topology on π1X,x.

Using their whiskers, rats have tactile capacities rivaling those of the human with our fingertips. We have carried out experiments to explore how neurons encode touch signals to build up a central representation. Touch signals begin with the receptors in the follicle of each whisker and can be traced to a columnar module in somatosensory cortex that is connected with the same whisker: the well-known whisker-to-column topography. We first examine the spatial organization or tactile information and find that the columnar topography acts as a framework for the organization of sensory information in behaving rats. Then, we consider the messages carried by neurons. Because rats can sweep their whiskers across surfaces to identify texture with very fine capacities, we ask what the brain representation of texture is. We will present evidence that the surface features of an object induce vibrations in the whisker shaft with distinct kinetic "signatures." The brain encodes the kinetics of whisker vibration to achieve a reliable representation of texture.

Enhancing the properties of dental resin composites is of interest to researchers. The objective of the present investigation was to improve the strength and fracture toughness of dental composites via addition of silicon carbide whiskers and substitution of commonly used filler materials with stabilized zirconia ceramic powder. It was also intended to study the effect of powder- to- whisker ratio on mechanical properties of the resultant composites. The flexural strength and fracture toughness of composite samples with different whiskers loadings were measured. It was found that addition of whiskers to the composites enhances the mechanical properties of the composites. The strength and fracture toughness increased by increasing the amount of whiskers. The flexural strength of a composite having 60wt% whisker and 10wt% zirconia powder was about 210 MPa while that of the composite having only 60wt% ceramic powder was about 110 MPa. The microstructural examinations revealed that reinforcing mechanism was whiskers pull-out as well as crack deflection.

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We define a new metric on the set of all closed linear operators between Hilbert spaces and investigate its properties. In particular, we show that the set of all closed operators with a closed range is an open subset of the set of all closed operators and the map T®T+ is an isometry in this metric. We also investigate the relationships between the topology induced by this metric and the gap metric.