Matching interior and exterior all-quadrilateral meshes with guaranteed angle bounds

This paper presents an extension of our earlier work on quadtree-based all-quadrilateral mesh generation, which generates guaranteed-quality meshes for the interior or exterior domain of any given planar curve. In this paper, we develop an algorithm to match the generated interior and exterior meshes with conformal boundary, preserving the guaranteed angle bounds. In addition, we introduce another automatic and robust approach to preserve sharp features. All the elements in the final mesh are within $[45^{\circ} - \varepsilon, 135^{\circ} + \varepsilon] (\varepsilon \leq 5^{\circ}),$ except small sharp angles present in the input geometry. Here, $\varepsilon$ is an input parameter. The smaller $\varepsilon$ is, the better angle bounds we can get. Finally, we group all the elements into six types, and most elements only need a few flops to construct the element stiffness matrix. This will significantly reduce the computational time during the finite element analysis.     

A System of Relational Syllogistic Incorporating Full Boolean Reasoning

We present a system of relational syllogistic, based on classical propositional logic, having primitives of the following form: $$\begin{array}{ll}\mathbf{Some}\, a \,{\rm are} \,R-{\rm related}\, {\rm to}\, \mathbf{some} \,b;\\ \mathbf{Some}\, a \,{\rm are}\,R-{\rm related}\, {\rm to}\, \mathbf{all}\, b;\\ \mathbf{All}\, a\, {\rm are}\,R-{\rm related}\, {\rm to}\, \mathbf{some}\, b;\\ \mathbf{All}\, a\, {\rm are}\,R-{\rm related}\, {\rm to}\, \mathbf{all} \,b.\end{array}$$ Such primitives formalize sentences from natural language like ‘All students read some textbooks’. Here a, b denote arbitrary sets (of objects), and R denotes an arbitrary binary relation between objects. The language of the logic contains only variables denoting sets, determining the class of set terms, and variables denoting binary relations between objects, determining the class of relational terms. Both classes of terms are closed under the standard Boolean operations. The set of relational terms is also closed under taking the converse of a relation. The results of the paper are the completeness theorem with respect to the intended semantics and the computational complexity of the satisfiability problem.     

Branching Time Logics \mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha } with Operations Until and Since Based on Bundles of Integer Numbers, Logical Consecutions, Deciding Algorithms

This paper is intended as an attempt to describe logical consequence in branching time logics. We study temporal branching time logics $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ which use the standard operations Until and Next and dual operations Since and Previous (LTL, as standard, uses only Until and Next). Temporal logics $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ are generated by semantics based on Kripke/Hinttikka structures with linear frames of integer numbers $\mathcal {Z}$ with a single node (glued zeros). For $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ , the permissible branching of the node is limited by α (where 1≤α≤ω). We prove that any logic $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ is decidable w.r.t. admissible consecutions (inference rules), i.e. we find an algorithm recognizing consecutions admissible in $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ . As a consequence, it implies that $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ itself is decidable and solves the satisfiability problem.     

A note on reverse scheduling with maximum lateness objective

The inverse and reverse counterparts of the single-machine scheduling problem $1||L_{\max }$ are studied in [2], in which the complexity classification is provided for various combinations of adjustable parameters (due dates and processing times) and for five different types of norm: $\ell _{1},\ell _{2},\ell _{\infty },\ell _{H}^{\Sigma } $ , and $\ell _{H}^{\max }$ . It appears that the $O(n^{2})$ -time algorithm for the reverse problem with adjustable due dates contains a flaw. In this note, we present the structural properties of the reverse model, establishing a link with the forward scheduling problem with due dates and deadlines. For the four norms $\ell _{1},\ell _{\infty },\ell _{H}^{\Sigma }$ , and $ \ell _{H}^{\max }$ , the complexity results are derived based on the properties of the corresponding forward problems, while the case of the norm $\ell _{2}$ is treated separately. As a by-product, we resolve an open question on the complexity of problem $1||\sum \alpha _{j}T_{j}^{2}$ .     

The addition of bounded quantification and partial functions to a computational logic and its theorem prover

We describe an extension to our quantifier-free computational logic to provide the expressive power and convenience of bounded quantifiers and partial functions. By quantifier we mean a formal construct which introduces a bound or indicial variable whose scope is some subexpression of the quantifier expression. A familiar quantifier is the Σ operator which sums the values of an expression over some range of values on the bound variable. Our method is to represent expressions of the logic as objects in the logic, to define an interpreter for such expressions as a function in the logic, and then define quantifiers as ‘mapping functions’. The novelty of our approach lies in the formalization of the interpreter and its interaction with the underlying logic. Our method has several advantages over other formal systems that provide quantifiers and partial functions in a logical setting. The most important advantage is that proofs not involving quantification or partial recursive functions are not complicated by such notions as ‘capturing’, ‘bottom’, or ‘continuity’. Naturally enough, our formalization of the partial functions is nonconstructive. The theorem prover for the logic has been modified to support these new features. We describe the modifications. The system has proved many theorems that could not previously be stated in our logic. Among them are:

? classic quantifier manipulation theorems, such as $$\sum\limits_{{\text{l}} = 0}^{\text{n}} {{\text{g}}({\text{l}}) + {\text{h(l) = }}} \sum\limits_{{\text{l = }}0}^{\text{n}} {{\text{g}}({\text{l}})} + \sum\limits_{{\text{l = }}0}^{\text{n}} {{\text{h(l)}};} $$

? elementary theorems involving quantifiers, such as the Binomial Theorem: $$(a + b)^{\text{n}} = \sum\limits_{{\text{l = }}0}^{\text{n}} {\left( {_{\text{i}}^{\text{n}} } \right)} \user2{ }{\text{a}}^{\text{l}} {\text{b}}^{{\text{n - l}}} ;$$

? elementary theorems about ‘mapping functions’ such as: $$(FOLDR\user2{ }'PLUS\user2{ O L) = }\sum\limits_{{\text{i}} \in {\text{L}}}^{} {{\text{i}};} $$

? termination properties of many partial recursive functions such as the fact that an application of the partial function described by $$\begin{gathered} (LEN X) \hfill \\ \Leftarrow \hfill \\ ({\rm I}F ({\rm E}QUAL X NIL) \hfill \\ {\rm O} \hfill \\ (ADD1 (LEN (CDR X)))) \hfill \\ \end{gathered} $$ terminates if and only if the argument ends in NIL;

? theorems about functions satisfying unusual recurrence equations such as the 91-function and the following list reverse function: $$\begin{gathered} (RV X) \hfill \\ \Leftarrow \hfill \\ ({\rm I}F (AND (LISTP X) (LISTP (CDR X))) \hfill \\ (CONS (CAR (RV (CDR X))) \hfill \\ (RV (CONS (CAR X) \hfill \\ (RV (CDR (RV (CDR X))))))) \hfill \\ X). \hfill \\ \end{gathered} $$

     

\lambda -Statistical convergence of fuzzy numbers and fuzzy functions of order \theta

In this paper, we introduce the concept of $\lambda $ -statistical convergence of order $\theta $ and strong $\lambda $ -summability of order $\theta $ for the sequence of fuzzy numbers. Further the same concept is extended to the sequence of fuzzy functions and introduce the spaces like $S_\lambda ^\theta (\hat{f})$ and $\omega _{\lambda p} ^\theta (\hat{f})$ . Some inclusion relations in those spaces and also the underlying relation between these two spaces are also obtained.     

On the class of λ-statistically convergent difference sequences of fuzzy numbers

In this study, we introduce the sets $\left[ V,\lambda ,p\right] _{\Updelta }^{{\mathcal{F}}},\left[ C,1,p\right] _{\Updelta }^{{\mathcal{F}}}$ and examine their relations with the classes of $ S_{\lambda }\left( \Updelta ,{\mathcal{F}}\right)$ and $ S_{\mu }\left( \Updelta ,{\mathcal{F}}\right)$ of sequences for the sequences $\left( \lambda _{n}\right)$ and $\left( \mu _{n}\right) , 0<p<\infty $ and difference sequences of fuzzy numbers.     

Representing Hyper-arithmetical Sets by Equations over Sets of Integers

Systems of equations with sets of integers as unknowns are considered. It is shown that the class of sets representable by unique solutions of equations using the operations of union and addition, defined as S+T={m+n?Om??S,n??T}, and with ultimately periodic constants is exactly the class of hyper-arithmetical sets. Equations using addition only can represent every hyper-arithmetical set under a simple encoding. All hyper-arithmetical sets can also be represented by equations over sets of natural numbers equipped with union, addition and subtraction $S \mathop {\mbox {$-^{\hspace {-.5em}\cdot }\,\,$}}T=\{m-n \mid m \in S, n \in T, m \geq n\}$ . Testing whether a given system has a solution is $\varSigma ^{1}_{1}$ -complete for each model. These results, in particular, settle the expressive power of the most general types of language equations, as well as equations over subsets of free groups.     

Integrated damping parameter and control design in structural systems for \bf{\mathcal{H}^2} and {\mathcal{H}^{\infty}} specifications

The paper presents a linear matrix inequality (LMI)-based approach for the simultaneous optimal design of output feedback control gains and damping parameters in structural systems with collocated actuators and sensors. The proposed integrated design is based on simplified $\mathcal{H}^2$ and $\mathcal{H}^{\infty}$ norm upper bound calculations for collocated structural systems. Using these upper bound results, the combined design of the damping parameters of the structural system and the output feedback controller to satisfy closed-loop $\mathcal{H}^2$ or $\mathcal{H}^{\infty}$ performance specifications is formulated as an LMI optimization problem with respect to the unknown damping coefficients and feedback gains. Numerical examples motivated from structural and aerospace engineering applications demonstrate the advantages and computational efficiency of the proposed technique for integrated structural and control design. The effectiveness of the proposed integrated design becomes apparent, especially in very large scale structural systems where the use of classical methods for solving Lyapunov and Riccati equations associated with $\mathcal{H}^2$ and $\mathcal{H}^{\infty}$ designs are time-consuming or intractable.     

Error Analysis of a Compact ADI Scheme for the 2D Fractional Subdiffusion Equation

In this paper, a Crank–Nicolson-type compact ADI scheme is proposed for solving two-dimensional fractional subdiffusion equation. The unique solvability, unconditional stability and convergence of the scheme are proved rigorously. Two error estimates are presented. One is $\mathcal{O }(\tau ^{\min \{2-\frac{\gamma }{2},\,2\gamma \}}+h_1^4+h^4_2)$ in standard $H^1$ norm, where $\tau $ is the temporal grid size and $h_1,h_2$ are spatial grid sizes; the other is $\mathcal{O }(\tau ^{2\gamma }+h_1^4+h^4_2)$ in $H^1_{\gamma }$ norm, a generalized norm which is associated with the Riemann–Liouville fractional integral operator. Numerical results are presented to support the theoretical analysis.     

Interpolation splines minimizing a semi-norm

Using S.L. Sobolev’s method, we construct the interpolation splines minimizing the semi-norm in $K_2(P_2)$ , where $K_2(P_2)$ is the space of functions $\phi $ such that $\phi ^{\prime } $ is absolutely continuous, $\phi ^{\prime \prime } $ belongs to $L_2(0,1)$ and $\int _0^1(\varphi ^{\prime \prime }(x)+\varphi (x))^2dx<\infty $ . Explicit formulas for coefficients of the interpolation splines are obtained. The resulting interpolation spline is exact for the trigonometric functions $\sin x$ and $\cos x$ . Finally, in a few numerical examples the qualities of the defined splines and $D^2$ -splines are compared. Furthermore, the relationship of the defined splines with an optimal quadrature formula is shown.     

Fast Polynomial-Space Algorithms Using Inclusion-Exclusion

Given a graph with n vertices, k terminals and positive integer weights not larger than c, we compute a minimum Steiner Tree in $\mathcal{O}^{\star}(2^{k}c)$ time and $\mathcal{O}^{\star}(c)$ space, where the $\mathcal{O}^{\star}$ notation omits terms bounded by a polynomial in the input-size. We obtain the result by defining a generalization of walks, called branching walks, and combining it with the Inclusion-Exclusion technique. Using this combination we also give $\mathcal{O}^{\star}(2^{n})$ -time polynomial space algorithms for Degree Constrained Spanning Tree, Maximum Internal Spanning Tree and #Spanning Forest with a given number of components. Furthermore, using related techniques, we also present new polynomial space algorithms for computing the Cover Polynomial of a graph, Convex Tree Coloring and counting the number of perfect matchings of a graph.     

Richardson Extrapolation on Some Recent Numerical Quadrature Formulas for Singular and Hypersingular Integrals and Its Study of Stability

Recently, we derived some new numerical quadrature formulas of trapezoidal rule type for the integrals \(I^{(1)}[g]=\int ^b_a \frac{g(x)}{x-t}\,dx\) and \(I^{(2)}[g]=\int ^b_a \frac{g(x)}{(x-t)^2}\,dx\) . These integrals are not defined in the regular sense; \(I^{(1)}[g]\) is defined in the sense of Cauchy Principal Value while \(I^{(2)}[g]\) is defined in the sense of Hadamard Finite Part. With \(h=(b-a)/n, \,n=1,2,\ldots \) , and \(t=a+kh\) for some \(k\in \{1,\ldots ,n-1\}, \,t\) being fixed, the numerical quadrature formulas \({Q}^{(1)}_n[g]\) for \(I^{(1)}[g]\) and \(Q^{(2)}_n[g]\) for \(I^{(2)}[g]\) are $$\begin{aligned} {Q}^{(1)}_n[g]=h\sum ^n_{j=1}f(a+jh-h/2),\quad f(x)=\frac{g(x)}{x-t}, \end{aligned}$$ and $$\begin{aligned} Q^{(2)}_n[g]=h\sum ^n_{j=1}f(a+jh-h/2)-\pi ^2g(t)h^{-1},\quad f(x)=\frac{g(x)}{(x-t)^2}. \end{aligned}$$ We provided a complete analysis of the errors in these formulas under the assumption that \(g\in C^\infty [a,b]\) . We actually show that $$\begin{aligned} I^{(k)}[g]-{Q}^{(k)}_n[g]\sim \sum ^\infty _{i=1} c^{(k)}_ih^{2i}\quad \text {as}\,n \rightarrow \infty , \end{aligned}$$ the constants \(c^{(k)}_i\) being independent of \(h\) . In this work, we apply the Richardson extrapolation to \({Q}^{(k)}_n[g]\) to obtain approximations of very high accuracy to \(I^{(k)}[g]\) . We also give a thorough analysis of convergence and numerical stability (in finite-precision arithmetic) for them. In our study of stability, we show that errors committed when computing the function \(g(x)\) , which form the main source of errors in the rest of the computation, propagate in a relatively mild fashion into the extrapolation table, and we quantify their rate of propagation. We confirm our conclusions via numerical examples.     

Reasoning about temporal properties of rational play

This article is about defining a suitable logic for expressing classical game theoretical notions. We define an extension of alternating-time temporal logic (ATL) that enables us to express various rationality assumptions of intelligent agents. Our proposal, the logic ATLP (ATL with plausibility) allows us to specify sets of rational strategy profiles in the object language, and reason about agents’ play if only these strategy profiles were allowed. For example, we may assume the agents to play only Nash equilibria, Pareto-optimal profiles or undominated strategies, and ask about the resulting behaviour (and outcomes) under such an assumption. The logic also gives rise to generalized versions of classical solution concepts through characterizing patterns of payoffs by suitably parameterized formulae of ATLP. We investigate the complexity of model checking ATLP for several classes of formulae: It ranges from $\Delta_{\mathbf{3}}^{\mathbf{P}}$ to PSPACE in the general case and from $\Delta_{\mathbf{3}}^{\mathbf{P}}$ to $\Delta_{\mathbf{4}}^{\mathbf{P}}$ for the most interesting subclasses, and roughly corresponds to solving extensive games with imperfect information.     

A transformation of series

The transformation (*) $$\sum\limits_{\nu = 0}^\infty {t_\nu z^\nu \to } \sum\limits_{\nu = 0}^\infty {\left\{ {\sum\limits_{\tau = 0}^{h - 1} {z^\tau } \Delta ^\nu t_{h\nu + \tau } + \frac{{z^h }}{{1 - z}}\Delta ^\nu t_{h(\nu + 1)} } \right\}} \left( {\frac{{z^{h + 1} }}{{1 - z}}} \right)^\nu$$ whereh≥0 is an integer and Δ operates upon the coefficients {t _v } of the series being transformed, is derived. Whenh=0, the above transformation is the generalised Euler transformation, of which (*) is itself a generalisation. Based upon the assumption that \(t_\nu = \int\limits_0^1 {\varrho ^\nu d\sigma (\varrho ) } (\nu = 0, 1,...)\) , where σ(?) is bounded and non-decreasing for 0≤?≤1 and subject to further restrictions, a convergence theory of (*) is given. Furthermore, the question as to when (*) functions as a convergence acceleration transformation is investigated. Also the optimal valne ofh to be taken is derived. A simple algorithm for constructing the partial sums of (*) is devised. Numerical illustrations relating to the case in whicht _v =(v+1) ^?1 (v=0,1,...) are given.     

Randomness Buys Depth for Approximate Counting

We show that the promise problem of distinguishing n-bit strings of relative Hamming weight \({1/2 + \Omega(1/{\rm lg}^{d-1} n)}\) from strings of weight \({1/2 - \Omega(1/{\rm \lg}^{d - 1} n)}\) can be solved by explicit, randomized (unbounded fan-in) poly(n)-size depth-d circuits with error \({\leq 1/3}\) , but cannot be solved by deterministic poly(n)-size depth-(d+1) circuits, for every \({d \geq 2}\) ; and the depth of both is tight. Our bounds match Ajtai’s simulation of randomized depth-d circuits by deterministic depth-(d + 2) circuits (Ann. Pure Appl. Logic; ’83) and provide an example where randomization buys resources. To rule out deterministic circuits, we combine Håstad’s switching lemma with an earlier depth-3 lower bound by the author (Computational Complexity 2009). To exhibit randomized circuits, we combine recent analyses by Amano (ICALP ’09) and Brody and Verbin (FOCS ’10) with derandomization. To make these circuits explicit, we construct a new, simple pseudorandom generator that fools tests \({A_1 \times A_2 \times \cdots \times A_{{\rm lg}{n}}}\) for \({A_i \subseteq [n], |A_{i}| = n/2}\) with error 1/n and seed length O(lg n), improving on the seed length \({\Omega({\rm lg}\, n\, {\rm lg}\, {\rm lg}\, n)}\) of previous constructions.     

Quantum image rotation by an arbitrary angle

In this paper, a novel method of quantum image rotation (QIR) based on shear transformations on NEQR quantum images is proposed. To compute the horizontal and vertical shear mappings required for rotation, we have designed quantum self-adder, quantum control multiplier, and quantum interpolation circuits as the basic computing units in the QIR implementation. Furthermore, we provide several examples of our results by presenting computer simulation experiments of QIR under \(30^\circ \), \(45^\circ \), and \(60^\circ \) rotation scenarios and have a discussion onto the anti-aliasing and computational complexity of the proposed QIR method.     

Model of embedded spaces: Peculiarities of dynamical embedding and geometry

The paper is devoted to construction of a geometry of 4D space-time arising from geometrization of classical electrodynamics and associated with the Model of Embedding Spaces (MES). The basic premise of the model is the assumption that each element of matter has an eigenspace, and the spacetime of the Universe is a result of mutual dynamic embedding of these spaces, such that the magnitude of partial embedding is determined by the relative Lagrangian density of the element interactions. It is shown that the required geometry of the MES is qualified as an inhomogeneous Finsler geometry (the metric is an arbitrary function of $\dot x$ ); the basic variant of the geometry (the unique law of parallel shift of metric and non-metric tensor quantities) is obtained as a differential generalization of the Riemannian geometry by the formal replacement ${\partial \mathord{\left/ {\vphantom {\partial {\partial x^i \mapsto }}} \right. \kern-0em} {\partial x^i \mapsto }}{\partial \mathord{\left/ {\vphantom {\partial {\partial x^i }}} \right. \kern-0em} {\partial x^i }} + {{2u^k \partial ^2 } \mathord{\left/ {\vphantom {{2u^k \partial ^2 } {\partial x^{\left[ i \right.} \partial u^{\left. k \right]} }}} \right. \kern-0em} {\partial x^{\left[ i \right.} \partial u^{\left. k \right]} }}$ , u = dx/ds; and the dynamic embedding provides linearity of this generalization. Additionally, some features ofMES cosmology, which can be caused by the natural electromagnetism of the Universe, are discussed.     

An efficient simulation algorithm on Kripke structures

A number of algorithms for computing the simulation preorder (and equivalence) on Kripke structures are available. Let $\varSigma $ denote the state space, ${\rightarrow }$ the transition relation and $P_{\mathrm {sim}}$ the partition of $\varSigma $ induced by simulation equivalence. While some algorithms are designed to reach the best space bounds, whose dominating additive term is $|P_{\mathrm {sim}}|^2$ , other algorithms are devised to attain the best time complexity $O(|P_{\mathrm {sim}}||{\rightarrow }|)$ . We present a novel simulation algorithm which is both space and time efficient: it runs in $O(|P_ {\mathrm {sim}}|^2 \log |P_{\mathrm {sim}}| + |\varSigma |\log |\varSigma |)$ space and $O(|P_{\mathrm {sim}}||{\rightarrow }|\log |\varSigma |)$ time. Our simulation algorithm thus reaches the best space bounds while closely approaching the best time complexity.