The modulus of the Fourier transform on a sphere determines 3-dimensional convex polytopes

Let 𝒫 and P ′ \mathcal{P}^{\prime} be 3-dimensional convex polytopes in R 3 \mathbb{R}^{3} and S ⊆ R 3 S\subseteq\mathbb{R}^{3} be a non-empty intersection of an open set with a sphere. As a consequence of a somewhat more general result it is proved that 𝒫 and P ′ \mathcal{P}^{\prime} coincide up to translation and/or reflection in a point if | ∫ P e - i ⁢ s ⋅ x ⁢ dx | = | ∫ P ′ e - i ⁢ s ⋅ x ⁢ dx | \bigl{\lvert}\int_{\mathcal{P}}e^{-i\mathbf{s}\cdot\mathbf{x}}\,\mathbf{dx}\bigr{\rvert}=\bigl{\lvert}\int_{\mathcal{P}^{\prime}}e^{-i\mathbf{s}\cdot\mathbf{x}}\,\mathbf{dx}\bigr{\rvert} for all s ∈ S \mathbf{s}\in S . This can be applied to the field of crystallography regarding the question whether a nanoparticle modelled as a convex polytope is uniquely determined by the intensities of its X-ray diffraction pattern on the Ewald sphere.

Comparing Degenerate Strings

Uncertain sequences are compact representations of sets of similar strings. They highlight common segments by collapsing them, and explicitly represent varying segments by listing all possible options. A generalized degenerate string (GD string) is a type of uncertain sequence. Formally, a GD string Ŝ is a sequence of n sets of strings of total size N, where the ith set contains strings of the same length ki but this length can vary between different sets. We denote by W the sum of these lengths k0, k1, …, kn-1. Our main result is an O(N + M)-time algorithm for deciding whether two GD strings of total sizes N and M, respectively, over an integer alphabet, have a non-empty intersection. This result is based on a combinatorial result of independent interest: although the intersection of two GD strings can be exponential in the total size of the two strings, it can be represented in linear space. We then apply our string comparison tool to devise a simple algorithm for computing all palindromes in Ŝ in O(min{W, n2}N)-time. We complement this upper bound by showing a similar conditional lower bound for computing maximal palindromes in Ŝ. We also show that a result, which is essentially the same as our string comparison linear-time algorithm, can be obtained by employing an automata-based approach.

Comparing Degenerate Strings

Uncertain sequences are compact representations of sets of similar strings. They highlight common segments by collapsing them, and explicitly represent varying segments by listing all possible options. A generalized degenerate string (GD string) is a type of uncertain sequence. Formally, a GD string Ŝ is a sequence of n sets of strings of total size N, where the ith set contains strings of the same length ki but this length can vary between different sets. We denote by W the sum of these lengths k0, k1, . . . , kn-1. Our main result is an 𝒪(N + M)-time algorithm for deciding whether two GD strings of total sizes N and M, respectively, over an integer alphabet, have a non-empty intersection. This result is based on a combinatorial result of independent interest: although the intersection of two GD strings can be exponential in the total size of the two strings, it can be represented in linear space. We then apply our string comparison tool to devise a simple algorithm for computing all palindromes in Ŝ in 𝒪(min{W, n2}N)-time. We complement this upper bound by showing a similar conditional lower bound for computing maximal palindromes in Ŝ. We also show that a result, which is essentially the same as our string comparison linear-time algorithm, can be obtained by employing an automata-based approach.

When Intersection Ideals of Graphs of Rings are a Divisor graph

Let R be a commutative principal ideal ring with unity. In this paper, we classify when the intersectiongraphs of ideals of a ring R G(R), is a divisor graph. We prove that the intersection graphs of ideals of a ring RG(R), is a divisor graph if and only if R is a local ring or it is a product of two local rings with each of them hasone chain of ideals. We also prove that G(R), is a divisor graph if it is a product of two local rings one of themhas at most two non-trivial ideals with empty intersection.

Structural Results for Conditionally Intersecting Families and some Applications

Let $k\ge d\ge 3$ be fixed. Let $\mathcal{F}$ be a $k$-uniform family on $[n]$. Then $\mathcal{F}$ is $(d,s)$-conditionally intersecting if it does not contain $d$ sets with union of size at most $s$ and empty intersection. Answering a question of Frankl, we present some structural results for families that are $(d,s)$-conditionally intersecting with $s\ge 2k+d-3$, and families that are $(k,2k)$-conditionally intersecting. As applications of our structural results we present some new proofs to the upper bounds for the size of the following $k$-uniform families on $[n]$: (a) $(d,2k+d-3)$-conditionally intersecting families with $n\ge 3k^5$; (b) $(k,2k)$-conditionally intersecting families with $n\ge k^2/(k-1)$; (c) Nonintersecting $(3,2k)$-conditionally intersecting families with $n\ge 3k\binom{2k}{k}$. Our results for $(c)$ confirms a conjecture of Mammoliti and Britz for the case $d=3$.

On the union of intersecting families

A family of sets is said to be intersecting if any two sets in the family have non-empty intersection. In 1973, Erdős raised the problem of determining the maximum possible size of a union of r different intersecting families of k-element subsets of an n-element set, for each triple of integers (n, k, r). We make progress on this problem, proving that for any fixed integer r ⩾ 2 and for any $$k \le ({1 \over 2} - o(1))n$$, if X is an n-element set, and $${\cal F} = {\cal F}_1 \cup {\cal F}_2 \cup \cdots \cup {\cal F}_r $$, where each $$ {\cal F}_i $$ is an intersecting family of k-element subsets of X, then $$|{\cal F}| \le \left( {\matrix{n \cr k \cr } } \right) - \left( {\matrix{{n - r} \cr k \cr } } \right)$$, with equality only if $${\cal F} = \{ S \subset X:|S| = k,\;S \cap R \ne \emptyset \} $$ for some R ⊂ X with |R| = r. This is best possible up to the size of the o(1) term, and improves a 1987 result of Frankl and Füredi, who obtained the same conclusion under the stronger hypothesis $$k < (3 - \sqrt 5 )n/2$$, in the case r = 2. Our proof utilizes an isoperimetric, influence-based method recently developed by Keller and the authors.

On Erdős–Ko–Rado for random hypergraphs I

A family of sets is intersecting if no two of its members are disjoint, and has the Erdős–Ko–Rado property (or is EKR) if each of its largest intersecting subfamilies has non-empty intersection.Denote by ${{\cal H}_k}(n,p)$ the random family in which each k-subset of {1, …, n} is present with probability p, independent of other choices. A question first studied by Balogh, Bohman and Mubayi asks: \begin{equation} {\rm{For what }}p = p(n,k){\rm{is}}{{\cal H}_k}(n,p){\rm{likely to be EKR}}? \end{equation} Here, for fixed c < 1/4, and $k \lt \sqrt {cn\log n} $ we give a precise answer to this question, characterizing those sequences p = p(n, k) for which \begin{equation} {\mathbb{P}}({{\cal H}_k}(n,p){\rm{is EKR}}{\kern 1pt} ) \to 1{\rm{as }}n \to \infty . \end{equation}

A problem of integer partitions and numerical semigroups

Let C be a set of positive integers. In this paper, we obtain an algorithm for computing all subsets A of positive integers which are minimals with the condition that if x1 + … + xn is a partition of an element in C, then at least a summand of this partition belongs to A. We use techniques of numerical semigroups to solve this problem because it is equivalent to give an algorithm that allows us to compute all the numerical semigroups which are maximals with the condition that has an empty intersection with the set C.