Analogous Problem Research Articles

Holomorphic one-forms, fibrations over the circle, and characteristic numbers of Kähler manifolds

AbstractWe prove that the only relation imposed on the Hodge and Chern numbers of a compact Kähler manifold by the existence of a nowhere zero holomorphic one-form is the vanishing of the Hirzebruch genus. We also treat the analogous problem for nowhere zero closed one-forms on smooth manifolds.

The dual Bonahon–Schläfli formula

Given a differentiable deformation of geometrically finite hyperbolic $3$-manifolds $(M_t)_t$, the Bonahon-Schl\"afli formula expresses the derivative of the volume of the convex cores $(C M_t)_t$ in terms of the variation of the geometry of its boundary, as the classical Schl\"afli formula does for the volume of hyperbolic polyhedra. Here we study the analogous problem for the dual volume, a notion that arises from the polarity relation between the hyperbolic space $\mathbb{H}^3$ and the de Sitter space $\mathrm{dS}^3$. The corresponding dual Bonahon-Schl\"afli formula has been originally deduced from Bonahon's work by Krasnov and Schlenker. Here, making use of the differential Schl\"afli formula and the properties of the dual volume, we give a (almost) self-contained proof of the dual Bonahon-Schl\"afli formula, without making use of Bonahon's original result.

An investigation of the use of specific problem-solving strategies by mathematics teachers in lessons

The aim of this study was to determine the strategies teachers offer to provide students with experience solving problems in the classroom. A total of 15 secondary school teachers serving in secondary schools were observed through 10 classes each, with no emphasis on the topics covered and grade levels taught. The researcher observing the classes performed a descriptive analysis of the problem-solving processes employed by the teachers through the lens of a series of known problem-solving strategies. The study revealed that the teachers employed a number of strategies as part of the problem-solving process. However, it was observed that the rate of application of strategies was rather low, even though specific strategies were nominally employed by the teachers. The study revealed that the strategy most frequently employed by the teachers during the problem-solving process was adopting a different point of view, while the strategy of making a drawing was also frequently applied. Moreover, the teachers employed the strategies of intelligent guessing and testing, working backwards, finding a pattern, solving a simpler analogous problem, and considering extreme cases, but with lesser frequency. On the other hand, the strategy of organizing data was not used by any of the teachers.

THE SOLUTION OF THE PROBLEM OF THE NON-STATIONARY HEAT CONDUCTION OF A TUNNEL UNDER CONDITIONS OF RADIAL VERIATION OF THERMAL PROPERTIES

The problem of building the temperature field over the heat exchange surface and in the surrounding rock mass of workings in conditions of constant and variable analogous problem has been solved by mathematical modeling. The results obtained by various methods show good agreement.

Word Embeddings Evaluation on Indonesian Translation of AI-Quran and Hadiths

Word vectors are an important part of machine learning. Word vectors are a numerical representation of text data. One of the methods that can be used to convert text into numerics is word embeddings. The word embeddings algorithm that researchers often use is Continuous Bag of Word, Skip-Gram, and FastText. This paper will discuss the transformation of textual data from Islamic knowledge domain documents into numerical forms using these three algorithms, then evaluate the word vector results using intrinsic and extrinsic evaluation techniques. We conduct intrinsic evaluations by determining the words to be evaluated, then checking for the existence of synonyms, antonyms, related words, and derived words from the nearest set of words based on vector values. We also tried to use vector words to solve word analogy problems. The best word vector in extrinsic evaluation is the result of the CBOW algorithm which is integrated with Binary Relevance and Multilayer Perceptron, with an accuracy value of 77.56% and a hamming loss value of 8.14%.

Minimum-Weight Combinatorial Structures Under Random Cost-Constraints

Recall that Janson showed that if the edges of the complete graph $K_n$ are assigned exponentially distributed independent random weights, then the expected length of a shortest path between a fixed pair of vertices is asymptotically equal to $(\log n)/n$. We consider analogous problems where edges have not only a random length but also a random cost, and we are interested in the length of the minimum-length structure whose total cost is less than some cost budget. For several classes of structures, we determine the correct minimum length structure as a function of the cost-budget, up to constant factors. Moreover, we achieve this even in the more general setting where the distribution of weights and costs are arbitrary, so long as the density $f(x)$ as $x\to 0$ behaves like $cx^\gamma$ for some $\gamma\geq 0$; previously, this case was not understood even in the absence of cost constraints. We also handle the case where each edge has several independent costs associated to it, and we must simultaneously satisfy budgets on each cost. In this case, we show that the minimum-length structure obtainable is essentially controlled by the product of the cost thresholds.

Counting independent sets in regular hypergraphs

Amongst d-regular r-uniform hypergraphs on n vertices, which ones have the largest number of independent sets? While the analogous problem for graphs (originally raised by Granville) is now well-understood, it is not even clear what the correct general conjecture ought to be; our goal here is to propose such a generalisation. Lending credence to our conjecture, we verify it within the class of ‘quasi-bipartite’ hypergraphs (a generalisation of bipartite graphs that seems natural in this context) by adopting the entropic approach of Kahn.

Performance of Azure-winged magpies in Aesop\u2019s fable paradigm

In this study, the improved Aesop’s fable paradigm—a series of experiments originally used to test whether some animals understand the causality associated with water replacement—was used to explore the cognitive ability of Azure-winged magpies (Cyanopica cyanus). Experimental results on causal cue tasks showed that the Azure-winged magpies prefer water-filled tubes over sand-filled tubes, heavy objects over light objects, and solid objects over hollow objects. However, they failed to notice the diameter and water level of the tubes. They also failed to pass the counterintuitive U-shaped tube task in arbitrary cue tasks. Our results demonstrated that Azure-winged magpies have a certain cognitive ability but not an understanding of causality, a characteristic comparable to that of other corvids. Moreover, Azure-winged magpies exhibited the ability of training transfer and analogical problem solving from the perspective of cognitive psychology. We believe that object-bias has little effect on Azure-winged magpies in this study. We can conclude that the Azure-winged magpies partially completed the tasks by trial-and-error learning.

Role of continuum in nuclear direct reactions with one-neutron halo nuclei: A one-dimensional model

We study the evolution of a single-particle wave function during the collision of a one dimensional potential well by another well, which can be regarded as a simple model for the problem of the scattering of a one-neutron halo nucleus by another nucleus. This constitutes an effective three-body problem, whose solution in three dimensions can be extremely complicated, particularly when breakup and rearrangement channels are to be considered. Our one-dimensional model provides the essential three-body nature of this problem, and allows for a much simpler application and assessment of different methods of solution. To simplify further the problem, we assume that the potential well representing the projectile moves according to a predetermined classical trajectory, although the internal motion of the "valence" particle is treated fully quantum-mechanically. This corresponds to a semiclassical approach of the scattering problem. Different approaches are investigated to understand the dynamics involving one-body halo-like systems: the "exact" time-dependent solution of the Schr\"odinger equation is compared to a numerical continuum-discretized coupled-channels (CC) calculation presenting various model cases including different reaction channels. This framework allows us to discuss the reaction mechanism and the role of continuum, whose inclusion in the CC calculation results to be crucial to reproduce the "exact" solution, even when the initial and final states are well bound. We also link each dynamical situation with analogous problem solved in a three dimensional (3D) CC framework, discussing the main challenges experienced in the usual 3D models.

ON THE EQUIVALENCE OF SOME CONVOLUTIONAL EQUALITIES IN SPACES OF SEQUENCES

The problem of the equivalence of two systems with $n$ convolutional equalities arose in investigation of the conditions of similarity in spaces of sequences of operators which are left inverse to the $n$-th degree of the generalized integration operator. In this paper we solve this problem. Note that we first prove the equivalence of two corresponding systems with $n$ equalities in the spaces of analytic functions, and then, using this statement, the main result of paper is obtained. Let $X$ be a vector space of sequences of complex numbers with K$\ddot{\rm o}$the normal topology from a wide class of spaces, ${\mathcal I}_{\alpha}$ be a generalized integration operator on $X$, $\ast$ be a nontrivial convolution for ${\mathcal I}_{\alpha}$ in $X$, and $(P_q)_{q=0}^{n-1}$ be a system of natural projectors with $\displaystyle x = \sum\limits_{q=0}^{n-1} P_q x$ for all $x\in X$. We established that a set $(a^{(j)})_{j=0}^{n-1}$ with $$ \max\limits_{0\le j \le n-1}\left\{\mathop{\overline{\lim}}\limits_{m\to\infty} \sqrt[m]{\left|\frac{a_{m}^{(j)}}{\alpha_m}\right|}\right\}<\infty $$ and a set $(b^{(j)})_{j=0}^{n-1}$ of elements of the space $X$ satisfy the system of equalities $$ b^{(j)}=a^{(j)}+\sum\limits_{k=0}^{n-1}({\mathcal I}_{\alpha}^{n-k-1} a^{(k)}) \ast {(P_{k}b^{(j)})}, \quad j = 0, 1, ... \, , \, n-1, $$ if and only if they satisfy the system of equalities $$ b^{(j)}=a^{(j)}+\sum\limits_{k=0}^{n-1}({\mathcal I}_{\alpha}^{n-k-1} b^{(k)}) \ast {(P_{k}a^{(j)})}, \quad j = 0, 1, ... \, , \, n-1. $$ Note that the assumption on the elements $(a^{(j)})_{j=0}^{n-1}$ of the space $X$ allows us to reduce the solution of this problem to the solution of an analogous problem in the space of functions analytic in a disc.

Stress state of an elastic rectangular domain under steady load

The dynamic elasticity problem for rectangular domain is presented at this paper. The conditions of ideal contact are given at the lateral sides and bottom edge of the domain; the upper edge is loaded by normal stress. Problem is formulated as the boundary value problem in steady-state case. It was necessary to find the wave field of the rectangular domain. The Fourier transform was applied to formulated problem and reduced it to one dimensional vector boundary problem. The last one was solved with the method based on matrix differential calculations which was successfully applied earlier to solve the analogous static problem Pozhylenkov O. V. (2019). According to this procedure of the solving the exact solution of the vector boundary problem was constructed in transform domain in the explicit form. Application of the inverse Fourier transform finalized the deriving of the stated problem solution. The analyses of wave field of rectangular domain depending on different load types, frequency values and domain size was done.

Exact Dual Bounds for Some Nonconvex Minimax Quadratic Optimization Problems

The nonconvex separable minimax quadratic optimization problem is analyzed. Two approaches to the problem solution are described, namely, by using SOCP relaxation and by using Lagrangian relaxation of a quadratic extremum analog problem. A condition is obtained that guarantees finding the value and the global extremum point of the problem of the considered class by calculating the dual bound of the equivalent quadratic extremum problem.

A generalization of the Escobar–Riemann mapping-type problem to smooth metric measure spaces

In this article, we introduce an analogous problem to Yamabe type problem considered by Case, J., which generalizes the Escobar-Riemann mapping problem for smooth metric measure spaces with boundary. The last problem will be called Escobar-Riemann mapping type problem. For this purpose, we consider the generalization of Sobolev Trace Inequality deduced by Bolley at. al. This trace inequality allows us to introduce an Escobar quotient and its infimum. This infimum we call the Escobar weighted constant. The Escobar-Riemann mapping type problem for smooth metric measure spaces in manifolds with boundary consists of finding a function which attains the Escobar weighted constant. Furthemore, we resolve the later problem when Escobar weighted constant is negative. Finally, we get an Aubin type inequality connecting the weighted Escobar constant for compact smooth metric measure space and the optimal constant for the trace inequality due to Bolley at. al.

Estimation Bounds of Beat Signal in the R-Mode Localization System

The R-Mode system is a terrestrial navigation system currently under development, which exploits existing means of medium frequency radio transmission. The positioning and timing performance depends on the estimation of the signals’ phase offset, from which the ranging information is derived. For an analogous problem such as the single-tone phase estimation, the Cramer-Rao bound (CRB) describes the minimal achievable performance in the mean squared error sense. For R-Mode, the problem involves the estimation of the phase offset for a beat signal, which can be described as the difference of phase estimation for the two aiding carriers next to the signal. This estimates are not statistically independent for finite observation, as we show in this paper. The effect becomes stronger for short observation times, which are important for a near real time application. In this contribution, we are interested in phase offset estimation for the signal models relevant to R-Mode: a beat signal and a beat signal combined with an MSK signal. A closed-form lower CRB is proposed for the aforementioned signal models phase estimation, as well as a generalization of the bound for the phase-difference estimation. Based on this derivation, optimized bit sequences are shown to improve performance of the estimates. The validity of the proposal is verified based on a simulation setup. Measurements acquired during a measurement campaign serve to further justify the usefulness of the bound. Some possible applications of such a bound are R-Mode coverage prediction and the associated phase estimators’ performance.

Some Methods of a Change in the Trickle Flow Structure during the Water Dispersion by Flat-Plate Jet Nozzles

Consideration is given to the possibility of the control of the structure of the trickle flow by using flat-plate jet nozzles for the elliptic shape of the surface of the spray zone. This research paper gives consideration to the data of experimental investigations of the water-air dispersed water during the internal mixture formation in the flat-plate nozzle as applied to the cooling of the ingot between the rollers in continuous steel casting machines. In this case, the water flow rate per nozzle and the water concentration can be reduced ten times. As a result, the ingot cooling intensity is reduced and the probability of the crack nucleation on the ingot surface is decreased. This research paper gives also the data of the experimental research carried out for a more efficient use of flat-plate nozzles when cooling the casting roller before its polishing and when heating it before charging the mill stand. It was shown that the intercrossing of trickle flows produced by two flat-plate jet nozzles arranged at an angle relative to each other results in the four-time increase in the spray zone surface area and the surface spray density is reduced two times. The analogous problem was solved using the kinetic energy of the disintegrating water film and the drops after these leave the flat-plate nozzle for the additional splitting when passing through the metal gauze. It turned out that the spray zone was increased threefold and the surface spray density was decreased two times. The trickle flow structure control options in question with the use of flat-plate nozzles contribute to the improved quality of the ingots, decreased water consumption, reduced number of the nozzles and their simplified arrangement on the collectors.

Evaluation of Vector Transformations for Russian Word2Vec and FastText Embeddings

Authors of Word2Vec claimed that their technology could solve the word analogy problem using the vector transformation in the introduced vector space. However, the practice demonstrates that it is not always true. In this paper, we investigate several Word2Vec and FastText model trained for the Russian language and find out reasons of such inconsistency. We found out that different types of words are demonstrating different behavior in the semantic space. FastText vectors are tending to find phonological analogies, while Word2Vec vectors are better in finding relations in geographical proper names. However, we found out that just four out of fifteen selected domains are demonstrating accuracy more that 0.8. We also draw a conclusion that in a common case, the task of word analogies could not be solved using a random word pair taken from two investigated categories. Our experiments have demonstrated that in some cases the length of the vectors could differ more than twice. Calculation of an average vector leads to a better solution here since it closer to more vectors.

Recursive Scheme for Angles of Random Simplices, and Applications to Random Polytopes

Consider a random simplex [X_1,ldots ,X_n] defined as the convex hull of independent identically distributed (i.i.d.) random points X_1,ldots ,X_n in mathbb {R}^{n-1} with the following beta density: Let J_{n,k}(beta ) be the expected internal angle of the simplex [X_1,ldots ,X_n] at its face [X_1,ldots ,X_k]. Define {tilde{J}}_{n,k}(beta ) analogously for i.i.d. random points distributed according to the beta' density {tilde{f}}_{n-1,beta } (x) propto (1+Vert xVert ^2)^{-beta }, xin mathbb {R}^{n-1}, beta > ({n-1})/{2}. We derive formulae for J_{n,k}(beta ) and {tilde{J}}_{n,k}(beta ) which make it possible to compute these quantities symbolically, in finitely many steps, for any integer or half-integer value of beta . For J_{n,1}(pm 1/2) we even provide explicit formulae in terms of products of Gamma functions. We give applications of these results to two seemingly unrelated problems of stochastic geometry: (i) We compute explicitly the expected f-vectors of the typical Poisson–Voronoi cells in dimensions up to 10. (ii) Consider the random polytope K_{n,d} := [U_1,ldots ,U_n] where U_1,ldots ,U_n are i.i.d. random points sampled uniformly inside some d-dimensional convex body K with smooth boundary and unit volume. Reitzner (Adv. Math. 191(1), 178–208 (2005)) proved the existence of the limit of the normalised expected f-vector of K_{n,d}: lim _{nrightarrow infty } n^{-{({d-1})/({d+1})}}{mathbb {E}}{mathbf {f}}(K_{n,d}) = {mathbf {c}}_d cdot Omega (K), where Omega (K) is the affine surface area of K, and {mathbf {c}}_d is an unknown vector not depending on K. We compute {mathbf {c}}_d explicitly in dimensions up to d=10 and also solve the analogous problem for random polytopes with vertices distributed uniformly on the sphere.

Partial data inverse problems and simultaneous recovery of boundary and coefficients for semilinear elliptic equations

We study various partial data inverse boundary value problems for the semilinear elliptic equation \Delta u+ a(x,u)=0 in a domain in \mathbb{R}^n by using the higher order linearization technique introduced by Lassas–Liimatainen–Lin–Salo and Feizmohammadi–Oksanen. We show that the Dirichlet-to-Neumann map of the above equation determines the Taylor series of a(x,z) at z = 0 under general assumptions on a(x,z) . The determination of the Taylor series can be done in parallel with the detection of an unknown cavity inside the domain or an unknown part of the boundary of the domain. The method relies on the solution of the linearized partial data Calderón problem by Ferreira–Kenig–Sjöstrand–Uhlmann, and implies the solution of partial data problems for certain semilinear equations \Delta u+ a(x,u) = 0 also proved by Krupchyk–Uhlmann. The results that we prove are in contrast to the analogous inverse problems for the linear Schrödinger equation. There recovering an unknown cavity (or part of the boundary) and the potential simultaneously are longstanding open problems, and the solution to the Calderón problem with partial data is known only in special cases when n \geq 3 .

Capture of a diffusive prey by multiple predators in confined space.

The first passage search of a diffusing target (prey) by multiple searchers (predators) in confinement is an important problem in the stochastic process literature. While the analogous problem in open space has been studied in some detail, a systematic study in confined space is still lacking. In this paper, we study the first passage times for this problem in one, two, and three dimensions. Due to confinement, the survival probability of the target takes a form ∼e^{-t/τ} at large times t. The characteristic capture timescale τ associated with the rare capture events are rather challenging to measure. We use a computational algorithm that allows us to estimate τ with high accuracy. We study in detail the behavior of τ as a function of the system parameters, namely, the number of searchers N, the relative diffusivity r of the target with respect to the searcher, and the system size. We find that τ deviates from the ∼1/N scaling seen in the case of a static target, and this deviation varies continuously with r and the spatial dimensions.

Statistical Physics for Medical Diagnostics: Learning, Inference, and Optimization Algorithms.

It is widely believed that cooperation between clinicians and machines may address many of the decisional fragilities intrinsic to current medical practice. However, the realization of this potential will require more precise definitions of disease states as well as their dynamics and interactions. A careful probabilistic examination of symptoms and signs, including the molecular profiles of the relevant biochemical networks, will often be required for building an unbiased and efficient diagnostic approach. Analogous problems have been studied for years by physicists extracting macroscopic states of various physical systems by examining microscopic elements and their interactions. These valuable experiences are now being extended to the medical field. From this perspective, we discuss how recent developments in statistical physics, machine learning and inference algorithms are coming together to improve current medical diagnostic approaches.