Initial Enlargement of Filtrations and Entropy of Poisson Compensators

Let μ be a Poisson random measure, let \mathbbF\mathbb{F} be the smallest filtration satisfying the usual conditions and containing the one generated by μ, and let \mathbbG\mathbb{G} be the initial enlargement of \mathbbF\mathbb{F} with the σ-field generated by a random variable G. In this paper, we first show that the mutual information between the enlarging random variable G and the σ-algebra generated by the Poisson random measure μ is equal to the expected relative entropy of the \mathbbG\mathbb{G}-compensator relative to the \mathbbF\mathbb{F}-compensator of the random measure μ. We then use this link to gain some insight into the changes of Doob–Meyer decompositions of stochastic processes when the filtration is enlarged from  \mathbbF\mathbb{F} to  \mathbbG\mathbb{G}. In particular, we show that if the mutual information between G and the σ-algebra generated by the Poisson random measure μ is finite, then every square-integrable \mathbbF\mathbb{F}-martingale is a \mathbbG\mathbb{G}-semimartingale that belongs to the normed space S¹\mathcal{S}^{1} relative to  \mathbbG\mathbb{G}.     

Affine extractors over prime fields

An affine extractor is a map that is balanced on every affine subspace of large enough dimension. We construct an explicit affine extractor AE from \mathbbFⁿ \mathbb{F}^n to \mathbbF\mathbb{F}, \mathbbF\mathbb{F} a prime field, so that AE(x) is exponentially close to uniform when x is chosen uniformly at random from an arbitrary affine subspace of \mathbbFⁿ \mathbb{F}^n of dimension at least δn, 0<δ≤1 a constant. Previously, Bourgain constructed such affine extractors when the size of \mathbbF\mathbb{F} is two. Our construction is in the spirit of but different than Bourgain’s construction. This allows for simpler analysis and better quantitative results.     

Mass equidistribution of Hilbert modular eigenforms

Let \mathbbF\mathbb{F} be a totally real number field, and let f traverse a sequence of non-dihedral holomorphic eigencuspforms on \operatornameGL₂/\mathbbF\operatorname{GL}_{2}/\mathbb{F} of weight (k₁,?,k_{[\mathbbF:\mathbbQ]})(k_{1},\ldots,k_{[\mathbb{F}:\mathbb{Q}]}), trivial central character and full level. We show that the mass of f equidistributes on the Hilbert modular variety as max(k₁,?,k_{[\mathbbF:\mathbbQ]}) ? ￥\max(k_{1},\ldots,k_{[\mathbb{F}:\mathbb{Q}]}) \rightarrow \infty.     

Some presentations of the real number field

It is proved that every two Σ-presentations of an ordered field \mathbbR \mathbb{R} of reals over \mathbbH\mathbbF ( \mathbbR ) \mathbb{H}\mathbb{F}\,\left( \mathbb{R} \right) , whose universes are subsets of \mathbbR \mathbb{R} , are mutually Σ-isomorphic. As a consequence, for a series of functions f:\mathbbR ? \mathbbR f:\mathbb{R} \to \mathbb{R} (e.g., exp, sin, cos, ln), it is stated that the structure \mathbbR \mathbb{R} = 〈R, +, ×, <, 0, 1, f〉 lacks such Σ-presentations over \mathbbH\mathbbF ( \mathbbR ) \mathbb{H}\mathbb{F}\,\left( \mathbb{R} \right) .     

A trade-off between collision probability and key size in universal hashing using polynomials

Let \mathbbF{\mathbb{F}} be a finite field and suppose that a single element of \mathbbF{\mathbb{F}} is used as an authenticator (or tag). Further, suppose that any message consists of at most L elements of \mathbbF{\mathbb{F}}. For this setting, usual polynomial based universal hashing achieves a collision bound of (L-1)/|\mathbbF|{(L-1)/|\mathbb{F}|} using a single element of \mathbbF{\mathbb{F}} as the key. The well-known multi-linear hashing achieves a collision bound of 1/|\mathbbF|{1/|\mathbb{F}|} using L elements of \mathbbF{\mathbb{F}} as the key. In this work, we present a new universal hash function which achieves a collision bound of mélog_m Lù/|\mathbbF|, m 3 2{m\lceil\log_m L\rceil/|\mathbb{F}|, m\geq 2}, using 1+élog_m Lù{1+\lceil\log_m L\rceil} elements of \mathbbF{\mathbb{F}} as the key. This provides a new trade-off between key size and collision probability for universal hash functions.     

Second cohomology group of extended W-algebras

Let \mathbbF\mathbb{F} be a field of characteristic 0, and let G be an additive subgroup of \mathbbF\mathbb{F}. We define a class of infinite-dimensional Lie algebras \mathbbF\mathbb{F}-basis {L _μ, V _μ, W _μ | μ ∈ G}, which are very closely related to W-algebras. In this paper, the second cohomology group of is determined.     

Cyclic codes over $${{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}$$

In this work, we focus on cyclic codes over the ring \mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}} , which is not a finite chain ring. We use ideas from group rings and works of AbuAlrub et.al. in (Des Codes Crypt 42:273–287, 2007) to characterize the ring (\mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂)/(xⁿ-1){({{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2})/(x^n-1)} and cyclic codes of odd length. Some good binary codes are obtained as the images of cyclic codes over \mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}} under two Gray maps that are defined. We also characterize the binary images of cyclic codes over \mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}} in general.     

Swan-like results for binomials and trinomials over finite fields of odd characteristic

Swan (Pac. J. Math. 12:1099–1106, 1962) gives conditions under which the trinomial x ⁿ + x ^k + 1 over \mathbbF₂{\mathbb{F}_{2}} is reducible. Vishne (Finite Fields Appl. 3:370–377, 1997) extends this result to trinomials over extensions of \mathbbF₂{\mathbb{F}_{2}}. In this work we determine the parity of the number of irreducible factors of all binomials and some trinomials over the finite field \mathbbF_q{\mathbb{F}_{q}}, where q is a power of an odd prime.     

Implementing the 4-dimensional GLV method on GLS elliptic curves with <Emphasis Type="Italic">j</Emphasis>-invariant 0

The Gallant–Lambert–Vanstone (GLV) method is a very efficient technique for accelerating point multiplication on elliptic curves with efficiently computable endomorphisms. Galbraith et al. (J Cryptol 24(3):446–469, 2011) showed that point multiplication exploiting the 2-dimensional GLV method on a large class of curves over \mathbbF_p²{\mathbb{F}_{p^2}} was faster than the standard method on general elliptic curves over \mathbbF_p{\mathbb{F}_{p}} , and left as an open problem to study the case of 4-dimensional GLV on special curves (e.g., j (E) = 0) over \mathbbF_p²{\mathbb{F}_{p^2}} . We study the above problem in this paper. We show how to get the 4-dimensional GLV decomposition with proper decomposed coefficients, and thus reduce the number of doublings for point multiplication on these curves to only a quarter. The resulting implementation shows that the 4-dimensional GLV method on a GLS curve runs in about 0.78 the time of the 2-dimensional GLV method on the same curve and in between 0.78 − 0.87 the time of the 2-dimensional GLV method using the standard method over \mathbbF_p{\mathbb{F}_{p}} . In particular, our implementation reduces by up to 27% the time of the previously fastest implementation of point multiplication on x86-64 processors due to Longa and Gebotys (CHES2010).     

Divisibility of polynomials over finite fields and combinatorial applications

Consider a maximum-length shift-register sequence generated by a primitive polynomial f over a finite field. The set of its subintervals is a linear code whose dual code is formed by all polynomials divisible by f. Since the minimum weight of dual codes is directly related to the strength of the corresponding orthogonal arrays, we can produce orthogonal arrays by studying divisibility of polynomials. Munemasa (Finite Fields Appl 4(3):252–260, 1998) uses trinomials over \mathbbF₂{\mathbb{F}_2} to construct orthogonal arrays of guaranteed strength 2 (and almost strength 3). That result was extended by Dewar et al. (Des Codes Cryptogr 45:1–17, 2007) to construct orthogonal arrays of guaranteed strength 3 by considering divisibility of trinomials by pentanomials over \mathbbF₂{\mathbb{F}_2} . Here we first simplify the requirement in Munemasa’s approach that the characteristic polynomial of the sequence must be primitive: we show that the method applies even to the much broader class of polynomials with no repeated roots. Then we give characterizations of divisibility for binomials and trinomials over \mathbbF₃{\mathbb{F}_3} . Some of our results apply to any finite field \mathbbF_q{\mathbb{F}_q} with q elements.     

Compact Surfaces with Constant Gaussian Curvature in Product Spaces

We prove that the only compact surfaces of positive constant Gaussian curvature in \mathbbH²×\mathbbR{\mathbb{H}^{2}\times\mathbb{R}} (resp. positive constant Gaussian curvature greater than 1 in \mathbbS²×\mathbbR{\mathbb{S}^{2}\times\mathbb{R}}) whose boundary Γ is contained in a slice of the ambient space and such that the surface intersects this slice at a constant angle along Γ, are the pieces of a rotational complete surface. We also obtain some area estimates for surfaces of positive constant Gaussian curvature in \mathbbH²×\mathbbR{\mathbb{H}^{2}\times\mathbb{R}} and positive constant Gaussian curvature greater than 1 in \mathbbS²×\mathbbR{\mathbb{S}^{2}\times\mathbb{R}} whose boundary is contained in a slice of the ambient space. These estimates are optimal in the sense that if the bounds are attained, the surface is again a piece of a rotational complete surface.     

A semilattice of numberings. II

\mathfrakc \mathfrak{c} -Universal semilattices \mathfrakA \mathfrak{A} of the power of the continuum (of an upper semilattice of m-degrees ) on admissible sets are studied. Moreover, it is shown that a semilattice of \mathbbH\mathbbF( \mathfrakM ) \mathbb{H}\mathbb{F}\left( \mathfrak{M} \right) -numberings of a finite set is \mathfrakc \mathfrak{c} -universal if \mathfrakM \mathfrak{M} is a countable model of a c-simple theory.     

Bijective maps on standard Borel subgroup of symplectic group preserving commutators

Let F be a field, and let \mathbbG\mathbb{G} be the standard Borel subgroup of the symplectic group Sp(2m, F). In this paper, we characterize the bijective maps ϕ: \mathbbG\mathbb{G} → \mathbbG\mathbb{G} satisfying ϕ[x, y] = [ϕ(x), ϕ(y)].     

Small resolutions and non-liftable Calabi-Yau threefolds

We use properties of small resolutions of the ordinary double point in dimension three to construct smooth non-liftable Calabi-Yau threefolds. In particular, we construct a smooth projective Calabi-Yau threefold over \mathbbF₃{\mathbb{F}_3} that does not lift to characteristic zero and a smooth projective Calabi-Yau threefold over \mathbbF₅{\mathbb{F}_5} having an obstructed deformation. We also construct many examples of smooth Calabi-Yau algebraic spaces over \mathbbF_p{\mathbb{F}_p} that do not lift to algebraic spaces in characteristic zero.     

The classification of irreducible admissible mod <Emphasis Type="Italic">p</Emphasis> representations of a <Emphasis Type="Italic">p</Emphasis>-adic GL<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

Let F be a finite extension of ℚ _p . Using the mod p Satake transform, we define what it means for an irreducible admissible smooth representation of an F-split p-adic reductive group over  [`( \mathbbF)]<sub>p</sub>\overline{ \mathbb{F}}_{p} to be supersingular. We then give the classification of irreducible admissible smooth GL <sub>n</sub> (F)-representations over  [`( \mathbbF)]_p\overline{ \mathbb{F}}_{p} in terms of supersingular representations. As a consequence we deduce that supersingular is the same as supercuspidal. These results generalise the work of Barthel–Livné for n=2. For general split reductive groups we obtain similar results under stronger hypotheses.     

Analytic cohomology groups in top degrees of Zariski open sets in {\mathbb{P}^n}

We show that if A is a closed analytic subset of \mathbbPⁿ{\mathbb{P}^n} of pure codimension q then Hⁱ(\mathbbPⁿ\ A,F){H^i(\mathbb{P}^n{\setminus} A,{\mathcal F})} are finite dimensional for every coherent algebraic sheaf F{{\mathcal F}} and every i 3 n-[\fracn-1q]{i\geq n-\left[\frac{n-1}{q}\right]} . If n-1 3 2q we show that H^n-2(\mathbbPⁿ\ A,F)=0{n-1\geq 2q\,{\rm we show that}\, H^{n-2}(\mathbb{P}^n{\setminus} A,{\mathcal F})=0} .     

$${\mathbb{Z}_2\mathbb{Z}_4}$$-linear codes: rank and kernel

A code C{{\mathcal C}} is \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C{{\mathcal C}} by deleting the coordinates outside X (respectively, Y) is a binary linear code (respectively, a quaternary linear code). The corresponding binary codes of \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-additive codes under an extended Gray map are called \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes. In this paper, the invariants for \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes, the rank and dimension of the kernel, are studied. Specifically, given the algebraic parameters of \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes, the possible values of these two invariants, giving lower and upper bounds, are established. For each possible rank r between these bounds, the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code with rank r is given. Equivalently, for each possible dimension of the kernel k, the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code with dimension of the kernel k is given. Finally, the bounds on the rank, once the kernel dimension is fixed, are established and the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code for each possible pair (r, k) is given.     

Dilation of Newton Polytope and <Emphasis Type="Italic">p</Emphasis>-adic Estimate

Let f(X) be a polynomial in n variables over the finite field  \mathbbF_q\mathbb{F}_{q}. Its Newton polytope Δ(f) is the convex closure in ℝ ⁿ of the origin and the exponent vectors (viewed as points in ℝ ⁿ ) of monomials in f(X). The minimal dilation of Δ(f) such that it contains at least one lattice point of $\mathbb{Z}_{>0}^{n}$\mathbb{Z}_{>0}^{n} plays a vital pole in the p-adic estimate of the number of zeros of f(X) in  \mathbbF_q\mathbb{F}_{q}. Using this fact, we obtain several tight and computational bounds for the dilation which unify and improve a number of previous results in this direction.     

The affine preservers of non-singular matrices

When \mathbbK{\mathbb{K}} is an arbitrary field, we study the affine automorphisms of M_n(\mathbbK){{\rm M}_n(\mathbb{K})} that stabilize GL_n(\mathbbK){{\rm GL}_n(\mathbb{K})}. Using a theorem of Dieudonné on maximal affine subspaces of singular matrices, this is easily reduced to the known case of linear preservers when n > 2 or # ${\mathbb{K} > 2}${\mathbb{K} > 2}. We include a short new proof of the more general Flanders theorem for affine subspaces of M_p,q(\mathbbK){{\rm M}_{p,q}(\mathbb{K})} with bounded rank. We also find that the group of affine transformations of M₂(\mathbbF₂){{\rm M}_2(\mathbb{F}_2)} that stabilize GL₂(\mathbbF₂){{\rm GL}_2(\mathbb{F}_2)} does not consist solely of linear maps. Using the theory of quadratic forms over \mathbbF₂{\mathbb{F}_2}, we construct explicit isomorphisms between it, the symplectic group Sp₄(\mathbbF₂){{\rm Sp}_4(\mathbb{F}_2)} and the symmetric group \mathfrakS₆{\mathfrak{S}_6}.     

Towards dimension expanders over finite fields

In this paper we study the problem of explicitly constructing a dimension expander raised by [3]: Let \mathbbFⁿ \mathbb{F}^n be the n dimensional linear space over the field \mathbbF\mathbb{F}. Find a small (ideally constant) set of linear transformations from \mathbbFⁿ \mathbb{F}^n to itself {A _i } _i∈I such that for every linear subspace V ⊂ \mathbbFⁿ \mathbb{F}^n of dimension dim(V)<n/2 we have