Seiberg–Witten Floer Homotopy Contact Invariant

We introduce a Floer homotopy version of the contact invariant introduced by Kronheimer–Mrowka–Ozsváth–Szabó. Moreover, we prove a gluing formula relating our invariant with the first author’s Bauer–Furuta type invariant, which refines Kronheimer–Mrowka’s invariant for 4-manifolds with contact boundary. As an application, we give a constraint for a certain class of symplectic fillings using equivariant KO-cohomology.

Upsilon Invariants from Cyclic Branched Covers

We extend the construction of Y-type invariants to null-homologous knots in rational homology three-spheres. By considering m-fold cyclic branched covers with m a prime power, this extension provides new knot concordance invariants of knots in S3. We give computations of some of these invariants for alternating knots and reprove independence results in the smooth concordance group.

A Corson Compact Space is Countable if the Complement of its Diagonal is Functionally Countable

A space X is called functionally countable if ƒ (X) is countable for any continuous function ƒ : X → Ø. Given an infinite cardinal k, we prove that a compact scattered space K with d(K) > k must have a convergent k+-sequence. This result implies that a Corson compact space K is countable if the space (K × K) \ ΔK is functionally countable; here ΔK = {(x, x): x ϵ K} is the diagonal of K. We also establish that, under Jensen’s Axiom ♦, there exists a compact hereditarily separable non-metrizable compact space X such that (X × X) \ ΔX is functionally countable and show in ZFC that there exists a non-separable σ-compact space X such that (X × X) \ ΔX is functionally countable.

Weighted Measures of Pseudorandom Binary Lattices

In cryptography one needs pseudorandom sequences whose short subsequences are also pseudorandom. To handle this problem, Dartyge, Gyarmati and Sárközy introduced weighted measures of pseudorandomness of binary sequences. In this paper we continue the research in this direction. We introduce weighted pseudorandom measure for multidimensional binary lattices and estimate weighted pseudorandom measure for truly random binary lattices. We also give lower bounds for weighted measures of even order and present an example by using the quadratic character of finite fields.

A Note on Generating a Power Basis over a Dedekind Ring

In this note, we show that the result [1, Proposition 5.2] is inaccurate. We further give and prove the correct modification of such a result. Some applications are also given.

Some Functional Upper Bounds for Fejér’s Sine Polynomial

We prove that for all integers n ≥ 1 and ɵ ≤ 8 ≤ π. This result refines inequalities due to Jackson (1911) and Turán (1938).

On Power Integral Bases for Certain Pure Number Fields Defined by x 36 − m

Let K = ℚ(α) be a number field generated by a complex root a of a monic irreducible polynomial ƒ (x) = x36 − m, with m ≠ ±1 a square free rational integer. In this paper, we prove that if m ≡ 2 or 3 (mod 4) and m ≠ ±1 (mod 9) then the number field K is monogenic. If m ≡ 1 (mod 4) or m ≡±1 (mod 9), then the number field K is not monogenic.

Real Hypersurfaces in ℂP 2 and ℂH 2 with Cyclic Parallel ∗-Ricci Tensor

In this paper, we prove that the ∗-Ricci tensor of a real hypersurface in complex projective plane ℂP 2 or complex hyperbolic plane ℂH 2 is cyclic parallel if and only if the hypersurface is of type (A). We find some three-dimensional real hypersurfaces having non-vanishing and non-parallel ∗-Ricci tensors which are cyclic parallel.

Heegaard Floer Homology, Degree-One Maps and Splicing Knot Complements

Suppose that K and K' are knots inside the homology spheres Y and Y', respectively. Let X = Y (K, K') be the 3-manifold obtained by splicing the complements of K and K' and Z be the three-manifold obtained by 0 surgery on K. When Y' is an L-space, we use the splicing formula of [1] to show that the rank of (X ) is bounded below by the rank of (Y ) if τ(K 2) = 0 and is bounded below by rank( (Z)) − 2 rank( (Y)) + 1 if τ(K') ≠ 0.

Criterion for the Coincidence of Strong and Weak Orlicz Spaces

We provide necessary and sufficient conditions for the coincidence, up to equivalence of the norms, between strong and weak Orlicz spaces. Roughly speaking, this coincidence holds true only for the so-called exponential spaces. We also find the exact value of the embedding constant which appears in the corresponding norm inequality.