Devising methods for planning a multifactorial multilevel experiment with high dimensionality

This paper considers the task of planning a multifactorial multilevel experiment for problems with high dimensionality. Planning an experiment is a combinatorial task. At the same time, the catastrophically rapid growth in the number of possible variants of experiment plans with an increase in the dimensionality of the problem excludes the possibility of solving it using accurate algorithms. On the other hand, approximate methods of finding the optimal plan have fundamental drawbacks. Of these, the main one is the lack of the capability to assess the proximity of the resulting solution to the optimal one. In these circumstances, searching for methods to obtain an accurate solution to the problem remains a relevant task. Two different approaches to obtaining the optimal plan for a multifactorial multilevel experiment have been considered. The first of these is based on the idea of decomposition. In this case, the initial problem with high dimensionality is reduced to a sequence of problems of smaller dimensionality, solving each of which is possible by using precise algorithms. The decomposition procedure, which is usually implemented empirically, in the considered problem of planning the experiment is solved by employing a strictly formally justified technique. The exact solutions to the problems obtained during the decomposition are combined into the desired solution to the original problem. The second approach directly leads to an accurate solution to the task of planning a multifactorial multilevel experiment for an important special case where the costs of implementing the experiment plan are proportional to the total number of single-level transitions performed by all factors. At the same time, it has been proven that the proposed procedure for forming a route that implements the experiment plan minimizes the total number of one-level changes in the values of factors. Examples of problem solving are given

The Classical and Quantum Photon Field for Non-compact Manifolds with Boundary and in Possibly Inhomogeneous Media

In this article I give a rigorous construction of the classical and quantum photon field on non-compact manifolds with boundary and in possibly inhomogeneous media. Such a construction is complicated by zero-modes that appear in the presence of non-trivial topology of the manifold or the boundary. An important special case is $${\mathbb {R}}^3$$ R 3 with obstacles. In this case the zero modes have a direct interpretation in terms of the topology of the obstacle. I give a formula for the renormalised stress energy tensor in terms of an integral kernel of an operator defined by spectral calculus of the Laplace Beltrami operator on differential forms with relative boundary conditions.

Nearest $$\varOmega $$-stable matrix via Riemannian optimization

We study the problem of finding the nearest $$\varOmega $$ Ω -stable matrix to a certain matrix A, i.e., the nearest matrix with all its eigenvalues in a prescribed closed set $$\varOmega $$ Ω . Distances are measured in the Frobenius norm. An important special case is finding the nearest Hurwitz or Schur stable matrix, which has applications in systems theory. We describe a reformulation of the task as an optimization problem on the Riemannian manifold of orthogonal (or unitary) matrices. The problem can then be solved using standard methods from the theory of Riemannian optimization. The resulting algorithm is remarkably fast on small-scale and medium-scale matrices, and returns directly a Schur factorization of the minimizer, sidestepping the numerical difficulties associated with eigenvalues with high multiplicity.

Rotational-monad metaphysics

The article presents a metaphysical concept, in the main tesa of which the simplest discrete element of physical reality is constituted — designated as a protomonad — which forms the basis of spatial forms of materiality, including space itself as such. The genesis of the protomonad is clarified by a certain way interpreted rotation of the spiritual essence, which in itself does not have an extension, and both the indicated essence and its rotation have a metaphysical order, and the dimension of physical space finds a rational interpretation through the characteristics of metaphysical rotation. The semantic aspects of complex mathematical constructs are considered that convey the semantics of rotations, and quite reasonably proposed by some mathematicians as the unified foundations of mathematics and physics, where the properties of constructs act as a mathematically strict co-proof of the validity of the concept. The meaning of number is explained as a method of restriction on infinite pre-physical multiplicity, and finite natural multiplicities are the result of such restrictions; the most important special case is the three-dimensional spatial metric given in the experiment, which appears as a restriction of an infinite-dimensional metaphysical space. The so-called principle of genetic inheritance is formulated, which makes it possible to remove the dialectical opposition between the one and the multiple and illustrates the categories of time and space as dialectical oppositions.

Exact simulation of Ornstein–Uhlenbeck tempered stable processes

There are two types of tempered stable (TS) based Ornstein–Uhlenbeck (OU) processes: (i) the OU-TS process, the OU process driven by a TS subordinator, and (ii) the TS-OU process, the OU process with TS marginal law. They have various applications in financial engineering and econometrics. In the literature, only the second type under the stationary assumption has an exact simulation algorithm. In this paper we develop a unified approach to exactly simulate both types without the stationary assumption. It is mainly based on the distributional decomposition of stochastic processes with the aid of an acceptance–rejection scheme. As the inverse Gaussian distribution is an important special case of TS distribution, we also provide tailored algorithms for the corresponding OU processes. Numerical experiments and tests are reported to demonstrate the accuracy and effectiveness of our algorithms, and some further extensions are also discussed.

Graph Theory Approach to Detect Examinees Involved in Test Collusion

Test collusion (TC) is sharing of test materials or answers to test questions before or during the test (important special case of TC is item preknowledge). Because of potentially large advantages for examinees involved, TC poses a serious threat to the validity of score interpretations. The proposed approach applies graph theory methodology to response similarity analyses for identifying groups of examinees involved in TC without using any knowledge about parts of test that were affected by TC. The approach supports different response similarity indices (specific to a particular type of TC) and different types of groups (connected components, cliques, or near-cliques). A comparison with an up-to-date method using real and simulated data is presented. Possible extensions and practical recommendations are given.

An axiomatic approach to existence and liveness for differential equations

This article presents an axiomatic approach for deductive verification of existence and liveness for ordinary differential equations (ODEs) with differential dynamic logic (dL). The approach yields proofs that the solution of a given ODE exists long enough to reach a given target region without leaving a given evolution domain. Numerous subtleties complicate the generalization of discrete liveness verification techniques, such as loop variants, to the continuous setting. For example, ODE solutions may blow up in finite time or their progress towards the goal may converge to zero. These subtleties are handled in dL by successively refining ODE liveness properties using ODE invariance properties which have a complete axiomatization. This approach is widely applicable: several liveness arguments from the literature are surveyed and derived as special instances of axiomatic refinement in dL. These derivations also correct several soundness errors in the surveyed literature, which further highlights the subtlety of ODE liveness reasoning and the utility of an axiomatic approach. An important special case of this approach deduces (global) existence properties of ODEs, which are a fundamental part of every ODE liveness argument. Thus, all generalizations of existence properties and their proofs immediately lead to corresponding generalizations of ODE liveness arguments. Overall, the resulting library of common refinement steps enables both the sound development and justification of new ODE existence and of liveness proof rules from dL axioms. These insights are put into practice through an implementation of ODE liveness proofs in the KeYmaera X theorem prover for hybrid systems.

Parameterized Approximation Algorithms for Bidirected Steiner Network Problems

The D irected S teiner N etwork (DSN) problem takes as input a directed graph G =( V , E ) with non-negative edge-weights and a set D ⊆ V × V of k demand pairs. The aim is to compute the cheapest network N⊆ G for which there is an s\rightarrow t path for each ( s , t )∈ D. It is known that this problem is notoriously hard, as there is no k 1/4− o (1) -approximation algorithm under Gap-ETH, even when parametrizing the runtime by k [Dinur & Manurangsi, ITCS 2018]. In light of this, we systematically study several special cases of DSN and determine their parameterized approximability for the parameter k . For the bi -DSNP lanar problem, the aim is to compute a solution N⊆ G whose cost is at most that of an optimum planar solution in a bidirected graph G , i.e., for every edge uv of G the reverse edge vu exists and has the same weight. This problem is a generalization of several well-studied special cases. Our main result is that this problem admits a parameterized approximation scheme (PAS) for k . We also prove that our result is tight in the sense that (a) the runtime of our PAS cannot be significantly improved, and (b) no PAS exists for any generalization of bi-DSNP lanar , under standard complexity assumptions. The techniques we use also imply a polynomial-sized approximate kernelization scheme (PSAKS). Additionally, we study several generalizations of bi -DSNP lanar and obtain upper and lower bounds on obtainable runtimes parameterized by k . One important special case of DSN is the S trongly C onnected S teiner S ubgraph (SCSS) problem, for which the solution network N⊆ G needs to strongly connect a given set of k terminals. It has been observed before that for SCSS a parameterized 2-approximation exists for parameter k [Chitnis et al., IPEC 2013]. We give a tight inapproximability result by showing that for k no parameterized (2 − ε)-approximation algorithm exists under Gap-ETH. Additionally, we show that when restricting the input of SCSS to bidirected graphs, the problem remains NP-hard but becomes FPT for k .

Eventual domination for linear evolution equations

We consider two $$C_0$$ C 0 -semigroups on function spaces or, more generally, Banach lattices and give necessary and sufficient conditions for the orbits of the first semigroup to dominate the orbits of the second semigroup for large times. As an important special case we consider an $$L^2$$ L 2 -space and self-adjoint operators A and B which generate $$C_0$$ C 0 -semigroups; in this situation we give criteria for the existence of a time $$t_1 \ge 0$$ t 1 ≥ 0 such that $$e^{tB} \ge e^{tA}$$ e tB ≥ e tA for all subsequent times $$t\ge t_1$$ t ≥ t 1 . As a consequence of our abstract theory, we obtain many surprising insights into the behaviour of various second and fourth order differential operators.

Competitive Agents in Certain and Uncertain Markets

This book uses concepts from optimization theory to develop an integrated analytic framework for treating consumer, producer, and market equilibrium analyses as special cases of a generic optimization problem. The same framework applies to both stochastic and non-stochastic decision settings, so that the latter is recognized as an (important) special case of the former. The analytic techniques are borrowed from convex analysis and variational analysis. Special emphasis is given to generalized notions of differentiability, conjugacy theory, and Fenchel's Duality Theorem. The book shows how virtually identical conjugate analyses form the basis for modeling economic behavior in each of the areas studied. The basic analytic concepts are borrowed from convex analysis. Special emphasis is given to generalized notions of differentiability, conjugacy theory, and Fenchel's Duality Theorem. It is demonstrated how virtually identical conjugate analyses form the basis for modelling economic behaviour in each of the areas studied.