https://www.chester.ac.uk/sites/files/chester/rep377.pdf

Analysis of Fractional Differential Equations

Differential Equations Driven by Moderately Irregular Signals.- The Signature of a Path.- Rough Paths.- Integration Along Rough Paths.- Differential Equations Driven by Rough Paths.

Differential Equations Driven by Rough Paths

The theory of error-correcting codes derived from curves in an algebraic geometry was initiated by the work of Goppa as generalizations of Bose-Chaudhuri-Hocquenghem (BCH), Reed-Solomon (RS), and Goppa codes. The development of the theory has received intense consideration since that time and the purpose of the paper is to review this work. Elements of the theory of algebraic curves, at a level sufficient to understand the code constructions and decoding algorithms, are introduced. Code constructions from particular classes of curves, including the Klein quartic, elliptic, and hyperelliptic curves, and Hermitian curves, are presented. Decoding algorithms for these classes of codes, and others, are considered. The construction of classes of asymptotically good codes using modular curves is also discussed.

/pdf/algebraic-geometry-codes-1izwm5be69.pdf

Algebraic-geometry codes

The concept of an error-correcting array gives a new bound on the minimum distance of linear codes and a decoding algorithm which decodes up to half this bound. This gives a unified point of view which explains several improvements on the minimum distance of algebraic-geometric codes. Moreover, it is explained in terms of linear algebra and the theory of semigroups only.

https://www.win.tue.nl/~ruudp/paper/20.pdf

The minimum distance of codes in an array coming from telescopic semigroups

A large number of inorganic oxides, mixed oxides, including alumina, silica, titania, zirconia, zeolites, carbon, clays and ion exchange resins have been employed as both supports and solid acid catalysts. Clay structure collapses at high temperatures and has to be stablilized. The thermal stability and pore size issues were addressed by pillaring of clays. Natural clays are acid treated or ion-exchanged to be used as solid acids, and the acidity and pore structure are dependent on the treatment methodology. Another important class of catalysts is the heteropoly acids (HPA), which are employed as homogeneous or heterogeneous catalysts, having both acid and redox properties. The focus of the current paper is to address the work done in our laboratory on the synergism between clays and heteropoly acids for the development of green processes. A comparison is also provided for the activity of these catalysts with other solid acids. Several alkylation, acylation, isomerization, condensation, dehydration, esterification, nitration and oxidation reactions which are useful in a wide spectrum of industries such as bulk, intermediates, dyes, plasticizer, pharmaceuticals, perfumes, flavours and other fine chemicals were investigated to improve the selectivity of the desired products.

Synergism of clay and heteropoly acids as nano-catalysts for the development of green processes with potential industrial applications

. We prove the following form of the Clemens conjecture in low degree. Let d ≤ 9, and let F be a general quintic threefold in P 4. Then (1) the Hilbert scheme of rational, smooth and irreducible curves of degree d on F is finite, nonempty, and reduced; moreover, each curve is embedded in F with normal bundle (−1) ⊕ (−1), and in P 4 with maximal rank. (2) On F, there are no rational, singular, reduced and irreducible curves of degree d, except for the 17,601,000 six-nodal plane quintics (found by Vainsencher). (3) On F, there are no connected, reduced and reducible curves of degree d with rational components.

Rational curves of degree at most 9 on a general quintic threefold

Introduction.- Surfaces in scrolls.- The Clifford index of smooth curves in |L| and the definition of the scrolls T(c, D, {D_{\lamda}}).- Two existence theorems.- The singular locus of the surface S' and the scroll T.- Postponed proofs.- Projective models in smooth scrolls.- Projective models in singular scrolls.- More on projective models in smooth scrolls of K3 surfaces of low Clifford-indices.- BN general and Clifford general K3 surfaces.- Projective models of K3 surfaces of low genus.- Some applications and open questions.- References.- Index.

K3 Projective models in scrolls

To each linear code $$C$$
 over a finite field we associate the matroid $$M(C)$$
 of its parity check matrix. For any matroid $$M$$
 one can define its generalized Hamming weights, and if a matroid is associated to such a parity check matrix, and thus of type $$M(C)$$
 , these weights are the same as those of the code $$C$$
 . In our main result we show how the weights $$d_1,\ldots ,d_k$$
 of a matroid $$M$$
 are determined by the $$\mathbb N $$
 -graded Betti numbers of the Stanley–Reisner ring of the simplicial complex whose faces are the independent sets of $$M$$
 , and derive some consequences. We also give examples which give negative results concerning other types of (global) Betti numbers, and using other examples we show that the generalized Hamming weights do not in general determine the $$\mathbb N $$
 -graded Betti numbers of the Stanley–Reisner ring. The negative examples all come from matroids of type $$M(C)$$
 .

/pdf/hamming-weights-and-betti-numbers-of-stanley-reisner-rings-13ilki5b2k.pdf

Hamming weights and betti numbers of stanley-reisner rings associated to matroids

We present several fundamental duality theorems for matroids and more general combinatorial structures. As a special case, these results show that the maximal cardinalities of fixed-ranked sets of a matroid determine the corresponding maximal cardinalities of the dual matroid. Our main results are applied to perfect matroid designs, graphs, transversals, and linear codes over division rings, in each case yielding a duality theorem for the respective class of objects.

/pdf/wei-type-duality-theorems-for-matroids-4a0bay2aip.pdf

Trygve Johnsen

Papers

Rational curves of degree at most 9 on a general quintic threefold

K3 Projective models in scrolls

Hamming weights and betti numbers of stanley-reisner rings associated to matroids

Wei-type duality theorems for matroids

A generalization of weight polynomials to matroids