A note on extensions of multilinear maps defined on multilinear varieties

Let $G_1, \ldots , G_k$ be finite-dimensional vector spaces over a prime field $\mathbb {F}_p$ . A multilinear variety of codimension at most $d$ is a subset of $G_1 \times \cdots \times G_k$ defined as the zero set of $d$ forms, each of which is multilinear on some subset of the coordinates. A map $\phi$ defined on a multilinear variety $B$ is multilinear if for each coordinate $c$ and all choices of $x_i \in G_i$ , $i\not =c$ , the restriction map $y \mapsto \phi (x_1, \ldots , x_{c-1}, y, x_{c+1}, \ldots , x_k)$ is linear where defined. In this note, we show that a multilinear map defined on a multilinear variety of codimension at most $d$ coincides on a multilinear variety of codimension $O_{k}(d^{O_{k}(1)})$ with a multilinear map defined on the whole of $G_1\times \cdots \times G_k$ . Additionally, in the case of general finite fields, we deduce similar (but slightly weaker) results.

Injective types in univalent mathematics

We investigate the injective types and the algebraically injective types in univalent mathematics, both in the absence and in the presence of propositional resizing. Injectivity is defined by the surjectivity of the restriction map along any embedding, and algebraic injectivity is defined by a given section of the restriction map along any embedding. Under propositional resizing axioms, the main results are easy to state: (1) Injectivity is equivalent to the propositional truncation of algebraic injectivity. (2) The algebraically injective types are precisely the retracts of exponential powers of universes. (2a) The algebraically injective sets are precisely the retracts of powersets. (2b) The algebraically injective (n+1)-types are precisely the retracts of exponential powers of universes of n-types. (3) The algebraically injective types are also precisely the retracts of algebras of the partial-map classifier. From (2) it follows that any universe is embedded as a retract of any larger universe. In the absence of propositional resizing, we have similar results that have subtler statements which need to keep track of universe levels rather explicitly, and are applied to get the results that require resizing.

A brief introduction to Quillen conjecture

Introduction: In 1971, Quillen stated a conjecture that on cohomology of arithmetic groups, a certain module structure over the Chern classes of the containing general linear group is free. Over time, many efforts has been dedicated into this conjecture. Some verified its correctness, some disproved it. So, the original Quillens conjecture is not correct. However, this conjecture still has great impacts on the field cohomology of group, especially on cohomology of arithmetic groups. This paper is meant to give a brief survey on Quillen conjecture and finally present a recent result that this conjecture has been verified by the authors. Method: In this work, we investigate the key reasons that makes Quillen conjecture fails. We review two of the reasons: 1) the injectivity of the restriction map; 2) the non-free of the image of the Quillen homomorphism. Those two reasons play important roles in the study of the correctness of Quillen conjecture. Results: In section 4, we present the cohomology of ring ​ which is isomorphic to the free module ​ over ​. This confirms the Quillen conjecture. Conclusion: The scope of the conjecture is not correct in Quillens original statement. It has been disproved in many examples and also been proved in many cases. Then determining the conjectures correct range of validity still in need. The result in section 4 is one of the confirmation of the validity of the conjecture.

Karakteristik Fragmen Gen Penyandi Cytochrome C Oxidase Subunit I (COI) Hama Kepik Penghisap Buah Kakao (Theobroma cacao L.)

The purpose of this research is to know the character of partial sequences of the COI gene form fruit-sucking pest. The gene fragments were isolated using PCR (Polymerase Chain Reaction) techniques with specific primers, HlpF and HlpR. Character of gene fragments observed were fragment size, nucleotide sequence, similiarity, restriction map, and hydrophobicity. The size of the fragment was determined by electrophoresis of PCR products, similarity values were determined by aligning the nucleotide sequence of the PCR product with the nucleotides present in GenBank, the restriction map determined by the RestrictionMapper program, and the hydrophobicity profile determined by the BioEdit program. The results showed that PCR yield fragment size 552 pb. The results of alignment analysis showed that PCR fragment had similarity of 88% with Helopeltis theivora, 87% with Helopeltis antonii, 87% with Helopeltis bradyi and 84% with Pacipeltis maesarum. Based on the results of the analysis using RestrictionMapper program shows partial sequences of the COI gene form fruit-sucking pest has 25 sites of restriction enzyme cutting which is class of type II endonuclease enzyme. The results of the hydrophobicity analysis using the BioEdit program indicate that the COI protein is hydrophilic and hydrophobic which shows the integrated COI protein on the membrane.Keywords: COI gene fragment, fruit-sucking pest, PCR, Cocoa Crop

S-paracompactness and S2-paracompactness

A topological space X is an S-paracompact if there exists a bijective function f from X onto a paracompact space Y such that for every separable subspace A of X the restriction map f|A from A onto f (A) is a homeomorphism. Moreover, if Y is Hausdorff, then X is called S2-paracompact. We investigate these two properties.

Traceless SU(2) representations of 2-stranded tangles

Given a 2-stranded tangle T contained in a ℤ-homology ball Y, we investigate the character variety R(Y, T) of conjugacy classes of traceless SU(2) representations of π1(Y \ T). In particular we completely determine the subspace of binary dihedral representations, and identify all of R(Y, T) for many tangles naturally associated to knots in S3. Moreover, we determine the image of the restriction map from R(T, Y) to the traceless SU(2) character variety of the 4-punctured 2-sphere (the pillowcase). We give examples to show this image can be non-linear in general, and show it is linear for tangles associated to pretzel knots.