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We compute a momentum space version of the entanglement spectrum and entanglement entropy of general Young tableau states, and one-point functions on Young tableau states. These physical quantities are used to measure the topology of the dual spacetime geometries in the context of gauge/gravity correspondence. The idea that Young tableau states can be obtained by superposing coherent states is explicitly verified. In this quantum superposition, a topologically distinct geometry is produced by superposing states dual to geometries with a trivial topology. Furthermore we have a refined bound for the overlap between coherent states and the rectangular Young tableau state, by using the techniques of symmetric groups and representations. This bound is exponentially suppressed by the total edge length of the Young tableau. It is also found that the norm squared of the overlaps is bounded above by inverse powers of the exponential of the entanglement entropies. We also compute the overlaps between Young tableau states and other states including squeezed states and multi-mode entangled states which have similarities with those appeared in quantum information theory.

In 1971, McMullen and Walkup posed the following conjecture, which is called the generalized lower bound conjecture: If$P$is a simplicial$d$-polytope then its$h$-vector ($h$_{0},$h$_{1}, …,$h$_{$d$}) satisfies $$ {h_0}\leq {h_1}\leq \ldots \leq {h_{{\left\lfloor {{d \left/ {2} \right.}} \right\rfloor }}} $$ . Moreover, if$h$_{$r$−1}=$h$_{$r$}for some $$ r\leq \frac{1}{2}d $$ then$P$can be triangulated without introducing simplices of dimension ≤$d$−$r$.

The rank of tensors is not additive with respect to the direct sum.

Szemerédi's regularity lemma and its variants are some of the most powerful tools in combinatorics. In this paper, we establish several results around the regularity lemma. First, we prove that whether or not we include the condition that the desired vertex partition in the regularity lemma is equitable has a minimal effect on the number of parts of the partition. Second, we use an algorithmic version of the (weak) Frieze–Kannan regularity lemma to give a substantially faster deterministic approximation algorithm for counting subgraphs in a graph. Previously, only an exponential dependence for the running time on the error parameter was known, and we improve it to a polynomial dependence. Third, we revisit the problem of finding an algorithmic regularity lemma, giving approximation algorithms for several co-NP-complete problems. We show how to use the weak Frieze–Kannan regularity lemma to approximate the regularity of a pair of vertex subsets. We also show how to quickly find, for each ε′>ε, an ε′-regular partition with k parts if there exists an ε-regular partition with k parts. Finally, we give a simple proof of the permutation regularity lemma which improves the tower-type bound on the number of parts in the previous proofs to a single exponential bound.

No abstract uploaded!

No abstract uploaded!

In 1975, P. Erdős proposed the problem of determining the maximum number $f(n)$ of edges in a graph of $n$ vertices in which any two cycles are of different lengths. In this paper, it is proved that $f(n)\geq n+32t-1$ for $t=27720r+169\ (r\geq 1)$ and $n\geq\frac{6911}{16}t^2+\frac{514441}{8}t-\frac{3309665}{16}$ . Consequently, $\liminf_{n\to\infty}\frac{f(n)-n}{\sqrt n}\geq\sqrt{2+\frac{2562}{6911}}$ .