搜索结果：Tower

The main result of this paper shows that a weak form of Tower Sealing holds in a generic extension of hod mice with a strong cardinal and a proper class of Woodin cardinals. We show Tower Sealing fails in such extensions in general. We show that this weak form of Tower Sealing (called Partial Tower Sealing) implies Sealing and that its consistency strength is below that of ZFC + there is a Woodin limit of Woodin cardinals.

A complete reduction $φ$ for derivatives in a differential field is a linear operator on the field over its constant subfield. The reduction enables us to decompose an element $f$ as the sum of a derivative and the remainder $φ(f)$. A direct application of $φ$ is that $f$ is in-field integrable if and only if $φ(f) = 0.$ In this paper, we present a complete reduction for derivatives in a primitive tower algorithmically. Typical examples for primitive towers are differential fields generated by (poly-)logarithmic functions and logarithmic integrals. Using remainders and residues, we provide a necessary and sufficient condition for an element from a primitive tower to have an elementary integral, and discuss how to construct telescopers for non-D-finite functions in some special primitive towers.

We consider the additive decomposition problem in primitive towers and present an algorithm to decompose a function in an S-primitive tower as a sum of a derivative in the tower and a remainder which is minimal in some sense. Special instances of S-primitive towers include differential fields generated by finitely many logarithmic functions and logarithmic integrals. A function in an S-primitive tower is integrable in the tower if and only if the remainder is equal to zero. The additive decomposition is achieved by viewing our towers not as a traditional chain of extension fields, but rather as a direct sum of certain subrings. Furthermore, we can determine whether or not a function in an S-primitive tower has an elementary integral without solving any differential equations. We also show that a kind of S-primitive towers, known as logarithmic towers, can be embedded into a particular extension where we can obtain a finer remainder.

A tower is a sequence of words alternating between two languages in such a way that every word is a subsequence of the following word. The height of the tower is the number of words in the sequence. If there is no infinite tower (a tower of infinite height), then the height of all towers between the languages is bounded. We study upper and lower bounds on the height of maximal finite towers with respect to the size of the NFA (the DFA) representation of the languages. We show that the upper bound is polynomial in the number of states and exponential in the size of the alphabet, and that it is asymptotically tight if the size of the alphabet is fixed. If the alphabet may grow, then, using an alphabet of size approximately the number of states of the automata, the lower bound on the height of towers is exponential with respect to that number. In this case, there is a gap between the lower and upper bound, and the asymptotically optimal bound remains an open problem. Since, in many cases, the constructed towers are sequences of prefixes, we also study towers of prefixes.

We study the energy spectrum of the composite magnetic monopole (CMM) in the model originally constructed in arXiv:1608.06951 to examine how the applicability of the weak gravity conjecture (WGC) propagates to lower energy scales in the effective field theory (EFT) framework. This is motivated by the WGC$=$distance conjecture (DC) connection suggested in the literature. Assuming that the limits of the model parameters correspond to the boundaries of the moduli space in a UV theory, we showed that the parameter dependence of the energy spectrum of the CMM is broadly in accord with the prediction of the WGC$=$DC connection. However, unlike the original DC, the low-energy excitations of the CMM are localized on the CMM. This leads us to propose an extension of the DC to include the localized tower of excited states. We discuss the implications of this extension of the DC to the swampland constraints on EFTs.

Theory and computations are provided for building of optimal (minimum weight) solid space towers (mast) up to one hundred kilometers in height. These towers can be used for tourism; scientific observation of space, observation of the Earth surface, weather and upper atmosphere experiment, and for radio, television, and communication transmissions. These towers can also be used to launch spaceships and Earth satellites. These macroprojects are not expensive. They require strong hard material (steel). Towers can be built using present technology. Towers can be used (for tourism, communication, etc.) during the construction process and provide self-financing for further construction. The tower design does not require human work at high altitudes; the tower is separated into sections; all construction can be done at the Earth surface. The transport system for a tower consists of a small engine (used only for friction compensation) located at the Earth surface. Problems involving security, control, repair, and stability of the proposed towers are addressed in other cited publications.

The main aim of this article is to study the topology of real Bott towers as special and interesting examples of real toric varieties. We first give a presentation of the fundamental group of a real Bott tower and show that the fundamental group is abelian if and only if the real Bott tower is a product of circles. We further prove that the fundamental group of a real Bott tower is always solvable and it is nilpotent if and only if it is abelian. We then describe the cohomology ring of a real Bott tower and also give recursive formulae for the Steifel Whitney classes. We derive combinatorial characterization for orientability of these manifolds and further give a combinatorial formula for the $(n-1)$th Steifel Whitney class. In particular, we show that if a Bott tower is orientable then the $(n-1)$th Steifel Whitney class must also vanish. Moreover, by deriving a combinatorial formula for the second Steifel-Whitney class we give a necessary and sufficient condition for the Bott tower to admit a spin structure. We finally prove the vanishing of all the Steifel-Whitney numbers and hence establish that these manifolds are null-cobordant.

A cyclic permutation $π:\{1, \dots, N\}\to \{1, \dots, N\}$ has a \emph{block structure} if there is a partition of $\{1, \dots, N\}$ into $k otin\{1,N\}$ segments (\emph{blocks}) permuted by $π$; call $k$ the \emph{period} of this block structure. Let $p_1<\dots <p_s$ be periods of all possible block structures on $π$. Call the finite string $(p_1/1,$ $p_2/p_1,$ $\dots,$ $p_s/p_{s-1}, N/p_s)$ the {\it renormalization tower of $π$}. The same terminology can be used for \emph{patterns}, i.e., for families of cycles of interval maps inducing the same (up to a flip) cyclic permutation. A renormalization tower $\mathcal M$ \emph{forces} a renormalization tower $\mathcal N$ if every continuous interval map with a cycle of pattern with renormalization tower $\mathcal M$ must have a cycle of pattern with renormalization tower $\mathcal N$. We completely characterize the forcing relation among renormalization towers. Take the following order among natural numbers: $ 4\gg 6\gg 3\gg \dots \gg 4n\gg 4n+2\gg 2n+1\gg\dots \gg 2\gg 1 $ understood in the strict sense. We show that the forcing relation among renormalization towers is given by the lexicographic extension of this order. Moreov

The automorphism tower of a group is obtained by computing its automorphism group, the automorphism group of THAT group, and so on, iterating transfinitely. Each group maps canonically into the next using inner automorphisms, and so at limit stages one can take a direct limit and continue the iteration. The tower is said to terminate if a fixed point is reached, that is, if a group is reached which is isomorphic to its automorphism group by the natural map. This occurs if a complete group is reached, one which is centerless and has only inner automorphisms. Wielandt [1939] proved the classical result that the automorphism tower of any centerless finite group terminates in finitely many steps. Rae and Roseblade [1970] proved that the automorphism tower of any centerless Cernikov group terminates in finitely many steps. Hulse [1970] proved that the the automorphism tower of any centerless polycyclic group terminates in countably many steps. Simon Thomas [1985] proved that the automorphism tower of any centerless group eventually terminates. In this paper, I remove the centerless assumption, and prove that every group has a terminating transfinite automorphism tower.

The Magnetic Tower of Hanoi puzzle - a modified "base 3" version of the classical Tower of Hanoi puzzle as described in earlier papers, is actually a small set of independent sister-puzzles, depending on the "pre-coloring" combination of the tower's posts. Starting with Red facing up on a Source post, working through an Intermediate - colored or Neutral post, and ending Blue facing up on a Destination post, we identify the different pre-coloring combinations in (S,I,D) order. The Tower's pre-coloring combinations are {[(R,B,B) / (R,R,B)] ; [(R,B,N) / (N,R,B)] ; [(N,B,N) / (N,R,N)] ; [R,N,B] ; [(R,N,N) / (N,N,B)] ; [N,N,N]}. In this paper we investigate these sister-puzzles, identify the algorithm that optimally solves each pre-colored puzzle, and prove its Optimality. As it turns out, five of the six algorithms, challenging on their own, are part of the algorithm solving the "natural", Free Magnetic Tower of Hanoi puzzle [N,N,N]. We start by showing that the N-disk Colored Tower [(R,B,B) / (R,R,B)] is solved by (3^N - 1)/2 moves. Defining "Algorithm Duration" as the ratio of number of algorithm-moves solving the puzzle to the number of algorithm-moves solving the Colored Tower, we

We examine the "homotopy coniveau tower" for a general cohomology theory on smooth k-schemes and give a new proof that the layers of this tower for K-theory agree with motivic cohomology. In addition, the homotopy coniveau tower agrees with Voevodsky's slice tower for $S^1$-spectra, giving a proof of a connectedness conjecture of Voevodsky. The homotopy coniveau tower construction extends to a tower of functors on the Morel-Voevodsky stable homotopy category, and we identify this $P^1$-stable homotopy coniveau tower with Voevodsky's slice filtration for $P^1$-spectra. We also show that the 0th layer for the motivic sphere spectrum is the motivic cohomology spectrum, which gives the layers for a general $P^1$-spectrum the structure of a module over motivic cohomology. This recovers and extends recent results of Voevodsky on the 0th layer of the slice filtration, and yields a spectral sequence that is reminiscent of the classical Atiyah-Hirzebruch spectral sequence.

Some well-known arithmetically Cohen-Macaulay configurations of linear varieties in $\mathbb{P}^r$ as $k$-configurations, partial intersections and star configurations are generalized by introducing tower schemes. Tower schemes are reduced schemes that are finite union of linear varieties whose support set is a suitable finite subset of $\mathbb{Z}_+^c$ called tower set. We prove that the tower schemes are arithmetically Cohen-Macaulay and we compute their Hilbert function in terms of their support. Afterwards, since even in codimension 2 not every arithmetically Cohen-Macaulay squarefree monomial ideal is the ideal of a tower scheme, we slightly extend this notion by defining generalized tower schemes (in codimension 2) and we show that the support of these configurations (the generalized tower set) gives a combinatorial characterization of the primary decomposition of the arithmetically Cohen-Macaulay squarefree monomial ideals.

The problem of the Hanoi Tower is a classic exercise in recursive programming: the solution has a simple recursive definition, and its complexity and the matching lower bound are the solution of a simple recursive function (the solution is so easy that most students memorize it and regurgitate it at exams without truly understanding it). We describe how some very minor changes in the rules of the Hanoi Tower yield various increases of complexity in the solution, so that they require a deeper analysis than the classical Hanoi Tower problem while still yielding exponential solutions. In particular, we analyze the problem fo the Bouncing Tower, where just changing the insertion and extraction position from the top to the middle of the tower results in a surprising increase of complexity in the solution: such a tower of $n$ disks can be optimally moved in $\sqrt{3}^n$ moves for $n$ even (i.e. less than a Hanoi Tower of same height), via $5$ recursive functions (or, equivalently, one recursion function with $5$ states).

In earlier work, we introduced the `Monster tower', a tower of fibrations associated to planar curves. We constructed an algorithm for classifying its points with respect to the equivalence relation generated by the action of the contact pseudogroup on the tower. Here, we construct the analogous tower for curves in $n$-space. (This tower is known as the Semple Bundle in Algebraic Geometry.) The pseudo-group of diffeomorphisms of $n$-space acts on each level of the extended tower. We take initial steps toward classifying points of this extended Monster tower under this pseudogroup action. Arnol'd's list of stable simple curve singularities plays a central role in these initial steps. We end with a list of open problems.

We investigate properties of stationary tower forcings and give conditions on stationary towers to derive the universally Baireness of sets of reals in $L(\mathbb{R})$.

A tower is a sequence of simplicial complexes connected by simplicial maps. We show how to compute a filtration, a sequence of nested simplicial complexes, with the same persistent barcode as the tower. Our approach is based on the coning strategy by Dey et al. (SoCG 2014). We show that a variant of this approach yields a filtration that is asymptotically only marginally larger than the tower and can be efficiently computed by a streaming algorithm, both in theory and in practice. Furthermore, we show that our approach can be combined with a streaming algorithm to compute the barcode of the tower via matrix reduction. The space complexity of the algorithm does not depend on the length of the tower, but the maximal size of any subcomplex within the tower.

Two languages are separable by a piecewise testable language if and only if there exists no infinite tower between them. An infinite tower is an infinite sequence of strings alternating between the two languages such that every string is a subsequence (scattered substring) of all the strings that follow. For regular languages represented by nondeterministic finite automata, the existence of an infinite tower is decidable in polynomial time. In this paper, we investigate the complexity of a particular method to compute a piecewise testable separator. We show that it is closely related to the height of maximal finite towers, and provide the upper and lower bounds with respect to the size of the given nondeterministic automata. Specifically, we show that the upper bound is polynomial with respect to the number of states with the cardinality of the alphabet in the exponent. Concerning the lower bound, we show that towers of exponential height with respect to the cardinality of the alphabet exist. Since these towers mostly turn out to be sequences of prefixes, we also provide a comparison with towers of prefixes.

In this thesis I study the automorphism tower of free nilpotent groups. Our main tool in studying the automorphism tower is to embed every group as a lattice in some Lie group. Using known rigidity results the automorphism group of the discrete group can be embedded into the automorphism group of the Lie group. This allows me to lift the description of the derivation tower of the free nilpotent Lie algebra to obtain information about the automorphism tower of the free nilpotent group. The main result in this thesis states that the automorphism tower of the free nilpotent group $Γ(n,d)$ on $n$ generators and nilpotency class $d$, stabilizes after finitely many steps. If the nilpotency class is small compared to the number of generators we have that the height of the automorphism tower is at most 3.

In this paper, we proved that there exist four distinct diffeomorphism classes of three-dimensional real Bott tower $M(A)=(S^1)^3/(\mathbb{Z}_2)^3$, and 12 distinct diffeomorphism classes of four-dimensional real Bott tower $M(A)=(S^1)^4/(\mathbb{Z}_2)^4$, where matrix $A$ corresponds to the action of $(\mathbb{Z}_2)^n$ on $(S^1)^n$ for n=3,4.