Hole dissections for planar figures

2021 ◽  
Vol 105 (563) ◽  
pp. 237-244
Author(s):  
Greg N. Frederickson
Keyword(s):  
Finite Number ◽  
Geometric Figure ◽  
Finite Set

A geometric dissection is a cutting of a geometric figure (or a finite set of figures) into pieces that we can rearrange to form another geometric figure (or finite set of figures). If our figures are required to be polygons, then there is always a dissection that has just a finite number of pieces. This was established by John Lowry [1], William Wallace [2], Farkas Bolyai [3], and Karl Gerwien [4]. The American Sam Loyd [5] and the Englishman Henry Ernest Dudeney [6, 7] emphasised the goal of minimising the number of pieces that resulted from such a standard dissection.

The inverse problem of recovering the coefficients of a differential equations on a graph

2020 ◽  
Vol 28 (5) ◽  
pp. 727-738
Author(s):  
Victor Sadovnichii ◽  
Yaudat Talgatovich Sultanaev ◽  
Azamat Akhtyamov
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Finite Number ◽  
Characteristic Equation ◽  
Arbitrary Number ◽  
New Class ◽  
Finite Set

AbstractWe consider a new class of inverse problems on the recovery of the coefficients of differential equations from a finite set of eigenvalues of a boundary value problem with unseparated boundary conditions. A finite number of eigenvalues is possible only for problems in which the roots of the characteristic equation are multiple. The article describes solutions to such a problem for equations of the second, third, and fourth orders on a graph with three, four, and five edges. The inverse problem with an arbitrary number of edges is solved similarly.

On Carleman Integral Operators

1970 ◽  
Vol 13 (3) ◽  
pp. 351-357
Author(s):  
Charles G. Costley
Keyword(s):  
Finite Number ◽  
Integral Operators ◽  
Finite Set ◽  
Limit Points ◽  
Image Position ◽  
Set Of Points

L2(a, b)1with the property2were originally defined by T. Carleman [4]. Here he imposed on the kernel the conditions of measurability and hermiticity,3for all x with the exception of a countable set with a finite number of limit points and4where Jδ denotes the interval [a, b] with the exception of subintervals |x - ξv| < δ; here ξv represents a finite set of points for which (3) fails to hold.

An Algorithmic Solution for a Word Problem in Group Theory

1964 ◽  
Vol 16 ◽  
pp. 509-516 ◽  
Author(s):  
N. S. Mendelsohn
Keyword(s):  
Group Theory ◽  
Finite Number ◽  
Word Problem ◽  
Finite Index ◽  
Unit Element ◽  
Systematic Procedure ◽  
Algorithmic Solution ◽  
Finite Set ◽  
The Right

This paper describes a systematic procedure which yields in a finite number of steps a solution to the following problem. Let G be a group generated by a finite set of generators g1, g2, g3, . . . , gr and defined by a finite set of relations R1 = R2 = . . . = Rk = I, where I is the unit element of G and R1R2, . . . , Rk are words in the gi and gi-1. Let H be a subgroup of G, known to be of finite index, and generated by a finite set of words, W1, W2, . . . , Wt. Let W be any word in G. Our problem is the following. Can we find a new set of generators for H, together with a set of representatives h1 = 1, h2, . . . , hu of the right cosets of H (i.e. G = H1 + Hh2 + . . . + Hhu) such that W can be expressed in the form W = Uhp, where U is a word in .

scholarly journals Estimates by gap potentials of free homotopy decompositions of critical Sobolev maps

2019 ◽  
Vol 9 (1) ◽  
pp. 1214-1250
Author(s):  
Jean Van Schaftingen
Keyword(s):  
Riemannian Manifold ◽  
Finite Number ◽  
Homotopy Group ◽  
Compact Riemannian Manifold ◽  
A Priori ◽  
Homotopy Classes ◽  
Continuous Map ◽  
Homotopy Decomposition ◽  
Sobolev Maps ◽  
Finite Set

Abstract A free homotopy decomposition of any continuous map from a compact Riemannian manifold 𝓜 to a compact Riemannian manifold 𝓝 into a finite number maps belonging to a finite set is constructed, in such a way that the number of maps in this free homotopy decomposition and the number of elements of the set to which they belong can be estimated a priori by the critical Sobolev energy of the map in Ws,p(𝓜, 𝓝), with sp = m = dim 𝓜. In particular, when the fundamental group π1(𝓝) acts trivially on the homotopy group πm(𝓝), the number of homotopy classes to which a map can belong can be estimated by its Sobolev energy. The estimates are particular cases of estimates under a boundedness assumption on gap potentials of the form $$\begin{array}{} \displaystyle \iint\limits_{\substack{(x, y) \in \mathcal{M} \times \mathcal{M} \\ d_\mathcal{N} (f (x), f (y)) \ge \varepsilon}} \frac{1}{d_\mathcal{M} (y, x)^{2 m}} \, \mathrm{d} y \, \mathrm{d}x. \end{array}$$ When m ≥ 2, the estimates scale optimally as ε → 0. When m = 1, the total variation of the maps appearing in the decomposition can be controlled by the gap potential. Linear estimates on the Hurewicz homomorphism and the induced cohomology homomorphism are also obtained.

scholarly journals Uncertainty principle for discrete Schrödinger evolution on graphs

2018 ◽  
Vol 123 (1) ◽  
pp. 51-71 ◽  
Author(s):  
Issac Álvarez-Romero
Keyword(s):  
Finite Number ◽  
Vector Space ◽  
Uncertainty Principle ◽  
The Matrix ◽  
Finite Set ◽  
Alpha Beta

We consider the Schrödinger evolution on graphs, i.e., solutions to the equation $\partial _t u(t,\alpha ) = i\sum _{\beta \in \mathcal {A}}L(\alpha ,\beta )u(t,\beta )$, where $\mathcal {A}$ is the set of vertices of the graph and the matrix $(L(\alpha ,\beta ))_{\alpha ,\beta \in \mathcal {A}}$ describes interaction between the vertices, in particular two vertices α and β are connected if $L(\alpha ,\beta )\neq 0$. We assume that the graph has a “web-like” structure, i.e., it consists of an inner part, formed by a finite number of vertices, and some threads attach to it.We prove that such a solution $u(t,\alpha )$ cannot decay too fast along one thread at two different times, unless it vanishes at this thread.We also give a characterization of the dimension of the vector space formed by all the solutions of $\partial _t u(t,\alpha ) = i\sum _{\beta \in \mathcal {A}}L(\alpha ,\beta )u(t,\beta )$, when $\mathcal {A}$ is a finite set, in terms of the number of the different eigenvalues of the matrix $L(\,\cdot \,,\,\cdot \,)$.

scholarly journals Axiomatic Characterizations of a Proportional Influence Measure for Sequential Projects with Imperfect Reliability

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 247
Author(s):  
Andries van van Beek ◽  
Peter Borm ◽  
Marieke Quant
Keyword(s):  
Finite Number ◽  
Task Group ◽  
Influence Measure ◽  
Axiomatic Characterizations ◽  
Finite Set ◽  
Group A ◽  
The Relationship

We define and axiomatically characterize a new proportional influence measure for sequential projects with imperfect reliability. We consider a model in which a finite set of players aims to complete a project, consisting of a finite number of tasks, which can only be carried out by certain specific players. Moreover, we assume the players to be imperfectly reliable, i.e., players are not guaranteed to carry out a task successfully. To determine which players are most important for the completion of a project, we use a proportional influence measure. This paper provides two characterizations of this influence measure. The most prominent property in the first characterization is task decomposability. This property describes the relationship between the influence measure of a project and the measures of influence one would obtain if one divides the tasks of the project over multiple independent smaller projects. Invariance under replacement is the most prominent property of the second characterization. If, in a certain task group, a specific player is replaced by a new player who was not in the original player set, this property states that this should have no effect on the allocated measure of influence of any other original player.

Post, Emil Leon (1897–1954)

2018 ◽  
Author(s):  
Michael Scanlan
Keyword(s):  
Finite Number ◽  
Propositional Logic ◽  
Decision Procedures ◽  
Truth Table ◽  
Formal Language Theory ◽  
Theory Of Computation ◽  
Symbol Manipulation ◽  
Canonical Systems ◽  
Finite Set ◽  
And Mathematics

Emil Post was a pioneer in the theory of computation, which investigates the solution of problems by algorithmic methods. An algorithmic method is a finite set of precisely defined elementary directions for solving a problem in a finite number of steps. More specifically, Post was interested in the existence of algorithmic decision procedures that eventually give a yes or no answer to a problem. For instance, in his dissertation, Post introduced the truth-table method for deciding whether or not a formula of propositional logic is a tautology. Post developed a notion of ‘canonical systems’ which was intended to encompass any algorithmic procedure for symbol manipulation. Using this notion, Post partially anticipated, in unpublished work, the results of Gödel, Church and Turing in the 1930s. This showed that many problems in logic and mathematics are algorithmically unsolvable. Post’s ideas influenced later research in logic, computer theory, formal language theory and other areas.

The false assumption underlying berry's paradox

1988 ◽  
Vol 53 (4) ◽  
pp. 1220-1223 ◽  
Author(s):  
James D. French
Keyword(s):  
Finite Number ◽  
Natural Number ◽  
English Language ◽  
Natural Numbers ◽  
Finite Set ◽  
False Assumption

This paper is divided into three sections. §1 consists of an argument against the validity of Berry's paradox; §2 consists of supporting arguments for the thesis presented in §1; and §3 examines the possibility of re-establishing the paradox.Berry's paradox, a semantic antinomy, is described on p. 4 of the textbook [4] as follows:For the sake of argument, let us admit that all the words of the English language are listed in some standard dictionary. Let T be the set of all thenatural numbers that can be described in fewer than twenty words of the English language. Since there are only a finite number of English words, there are only finitely many combinations of fewer than twenty such words—that is, T is a finite set. Quite obviously, then, there are natural numbers which are greater than all the elements of T; hence there is a least natural number which cannot be described in fewer than twenty words of the English language. By definition, this number is not in T; yet we have described it in sixteen words, hence it is in T.We are faced with a glaring contradiction; since the above argument would be unimpeachable if we admitted the existence of the set T, we are irrevocably led to the conclusion that a set such as T simply cannot exist.

scholarly journals Time complexity of algorithms of full analysis of dynamical systems with discrete time, defined on a finite set of states

2016 ◽  
Vol 24 ◽  
pp. 95
Author(s):  
V.I. Ruban ◽  
A.A. Rudenko
Keyword(s):  
Dynamical Systems ◽  
Finite Number ◽  
Discrete Time ◽  
Time Complexity ◽  
Discrete Dynamical Systems ◽  
Complexity Of Algorithms ◽  
Finite Set ◽  
Full Analysis

For discrete dynamical systems with a finite number of states, we obtain order of time complexity ascension of algorithms of their full analysis.

scholarly journals A Generalization of Partition Identities for First Differences of Partitions of $n$ Into at Most $m$ Parts

10.37236/8199 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Acadia Larsen
Keyword(s):  
Finite Number ◽  
Linear Combination ◽  
Prime Power ◽  
Power Number ◽  
Partition Identities ◽  
Partition Identity ◽  
Combinatorial Interpretations ◽  
Finite Set

We show for a prime power number of parts $m$ that the first differences of partitions into at most $m$ parts can be expressed as a non-negative linear combination of partitions into at most $m-1$ parts. To show this relationship, we combine a quasipolynomial construction of $p(n,m)$ with a new partition identity for a finite number of parts. We prove these results by providing combinatorial interpretations of the quasipolynomial of $p(n,m)$ and the new partition identity.  We extend these results by establishing conditions for when partitions of $n$ with parts coming from a finite set $A$ can be expressed as a non-negative linear combination of partitions with parts coming from a finite set $B$.

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