Lower Bounds for RAMs and Quantifier Elimination

We are considering RAMs $N_{n}$, with wordlength $n=2^{d}$, whose arithmetic instructions are the arithmetic operations multiplication and addition modulo $2^{n}$, the unary function $\min\lbrace 2^{x}, 2^{n}-1\rbrace$, the binary functions $\lfloor x/y\rfloor$ (with $\lfloor x/0 \rfloor =0$), $\max(x,y)$, $\min(x,y)$, and the boolean vector operations $\wedge,\vee,\neg$ defined on $0,1$ sequences of length $n$. It also has the other RAM instructions. The size of the memory is restricted only by the address space, that is, it is $2^{n}$ words. The RAMs has a finite instruction set, each instruction is encoded by a fixed natural number independently of $n$. Therefore a program $P$ can run on each machine $N_{n}$, if $n=2^{d}$ is sufficiently large. We show that there exists an $\epsilon>0$ and a program $P$, such that it satisfies the following two conditions. (i) For all sufficiently large $n=2^{d}$, if $P$ running on $N_{n}$ gets an input consisting of two words $a$ and $b$, then, in constant time, it gives a $0,1$ output $P_{n}(a,b)$. (ii) Suppose that $Q$ is a program such that for each sufficiently large $n=2^{d}$, if $Q$, running on $N_{n}$, gets a word $a$ of length $n$ as an input, then it decides whether there exists a word $b$ of length $n$ such that $P_{n}(a,b)=0$. Then, for infinitely many positive integers $d$, there exists a word $a$ of length $n=2^{d}$, such that the running time of $Q$ on $N_{n}$ at input $a$ is at least $\epsilon (\log d)^{\frac{1}{2}} (\log \log d)^{-1}$

A better upper bound on the number of triangulations of a planar point set

We show that a point set of cardinality $n$ in the plane cannot be the vertex set of more than $59^n O(n^{-6})$ straight-edge triangulations of its convex hull. This improves the previous upper bound of $276.75^n$.Comment: 6 pages, 1 figur

Almost maximally almost-periodic group topologies determined by T-sequences

A sequence $\{a_n\}$ in a group $G$ is a {\em $T$-sequence} if there is a Hausdorff group topology $\tau$ on $G$ such that $a_n\stackrel\tau\longrightarrow 0$. In this paper, we provide several sufficient conditions for a sequence in an abelian group to be a $T$-sequence, and investigate special sequences in the Pr\"ufer groups $\mathbb{Z}(p^\infty)$. We show that for $p\neq 2$, there is a Hausdorff group topology $\tau$ on $\mathbb{Z}(p^\infty)$ that is determined by a $T$-sequence, which is close to being maximally almost-periodic--in other words, the von Neumann radical $\mathbf{n}(\mathbb{Z}(p^\infty),\tau)$ is a non-trivial finite subgroup. In particular, $\mathbf{n}(\mathbf{n}(\mathbb{Z}(p^\infty),\tau)) \subsetneq \mathbf{n}(\mathbb{Z}(p^\infty),\tau)$. We also prove that the direct sum of any infinite family of finite abelian groups admits a group topology determined by a $T$-sequence with non-trivial finite von Neumann radical.Comment: v2 - accepted (discussion on non-abelian case is removed, replaced by new results on direct sums of finite abelian groups

Datalog vs first-order logic

Our main result is that every datalog query expressible in first-order logic is bounded; in terms of classical model theory it is a kind of compactness theorem for finite structures. In addition, we give some counter-examples delimiting the main result

A new proof of the graph removal lemma

Let H be a fixed graph with h vertices. The graph removal lemma states that every graph on n vertices with o(n^h) copies of H can be made H-free by removing o(n^2) edges. We give a new proof which avoids Szemer\'edi's regularity lemma and gives a better bound. This approach also works to give improved bounds for the directed and multicolored analogues of the graph removal lemma. This answers questions of Alon and Gowers.Comment: 17 page

Lower bounds for on-line graph colorings

We propose two strategies for Presenter in on-line graph coloring games. The first one constructs bipartite graphs and forces any on-line coloring algorithm to use $2\log_2 n - 10$ colors, where $n$ is the number of vertices in the constructed graph. This is best possible up to an additive constant. The second strategy constructs graphs that contain neither $C_3$ nor $C_5$ as a subgraph and forces $\Omega(\frac{n}{\log n}^\frac{1}{3})$ colors. The best known on-line coloring algorithm for these graphs uses $O(n^{\frac{1}{2}})$ colors

On graphs with a large chromatic number containing no small odd cycles

In this paper, we present the lower bounds for the number of vertices in a graph with a large chromatic number containing no small odd cycles

Sorting and Selection with Imprecise Comparisons

In experimental psychology, the method of paired comparisons was proposed as a means for ranking preferences amongst n elements of a human subject. The method requires performing all (n2) comparisons then sorting elements according to the number of wins. The large number of comparisons is performed to counter the potentially faulty decision-making of the human subject, who acts as an imprecise comparator. We consider a simple model of the imprecise comparisons: there exists some δ> 0 such that when a subject is given two elements to compare, if the values of those elements (as perceived by the subject) differ by at least δ, then the comparison will be made correctly; when the two elements have values that are within δ, the outcome of the comparison is unpredictable. This δ corresponds to the just noticeable difference unit (JND) or difference threshold in the psychophysics literature, but does not require the statistical assumptions used to define this value. In this model, the standard method of paired comparisons minimizes the errors introduced by the imprecise comparisons at the cost of (n2) comparisons. We show that the same optimal guarantees can be achieved using 4 n 3/2 comparisons, and we prove the optimality of our method. We then explore the general tradeoff between the guarantees on the error that can be made and number of comparisons for the problems of sorting, max-finding, and selection. Our results provide close-to-optimal solutions for each of these problems.Engineering and Applied Science

Characteristics of Glial Reaction in the Perinatal Rat Cortex: Effect of Lesion Size in the ‘Critical Period’

In this study we investigate the capability of lesions, performed between embryonic day E18 and postnatal day P6, to provoke glial reaction. Two different lesion types were applied: ‘severe’ lesion (tissue defect) and ’light’ lesion (stab wound). The glial reaction was detected with immunostain[ng against glial fibrillary acidic protein. When performed as early as P0, severe lesions could result in reactive gliosis, which persisted even after a month. The glial reaction was detected at P6/P7 and became strong by P8, regardless of the age when the animals were lesioned between P0 and P5. Namely, a strict limit could be estimated for the age when reactive glia were already found rather than for the age when glial reaction-provoking lesions could occur. After prenatal lesions, no glial reaction developed, but the usual glia limitans covered the deformed brain, surface. Light lesions provoked glial reactions when performed at P6. In conclusion, three scenarios were found, depending on the age of the animal at injury: (i) healing without glial reaction, regardless of the remaining deformation; (ii) depending on the size of the lesion, either healing without residuum or with remaining tissue defect plus reactive gliosis; and (iii) healing always with reactive gliosis. The age limits between them were at P0 and P5. The glial reactivity seemingly appears after the end of the neuronal migration and just precedes the massive transformation of the radial glia into astrocytes. Estimating the position of the appearance of glial reactivity among the events of cortical maturation can help to adapt the experimental results to humans