Sets Characterized by Missing Sums and Differences

A more sums than differences (MSTD) set is a finite subset S of the integers such |S+S| > |S-S|. We show that the probability that a uniform random subset of {0, 1, ..., n} is an MSTD set approaches some limit rho > 4.28 x 10^{-4}. This improves the previous result of Martin and O'Bryant that there is a lower limit of at least 2 x 10^{-7}. Monte Carlo experiments suggest that rho \approx 4.5 \x 10^{-4}. We present a deterministic algorithm that can compute rho up to arbitrary precision. We also describe the structure of a random MSTD subset S of {0, 1, ..., n}. We formalize the intuition that fringe elements are most significant, while middle elements are nearly unrestricted. For instance, the probability that any ``middle'' element is in S approaches 1/2 as n -> infinity, confirming a conjecture of Miller, Orosz, and Scheinerman. In general, our results work for any specification on the number of missing sums and the number of missing differences of S, with MSTD sets being a special case.Comment: 32 pages, 1 figure, 1 tabl

Information about the Zasechnaya Line in the Lower Part of the Volga in the Astrakhan Region (from archival sources and materials of archaeological studies)

The article considers the Moscow government activities aimed at ensuring the protection of the southern borders of the state, located in the lower part of the Volga River. The given analysis of written sources shows the scale of the works on creation of defensive objects in the XVI – XVIII centuries, included in the state Zasechnaya line, protecting the lands of the Tsardom of Russia from the raids of nomads. A list and a brief description of military fortresses built in the Astrakhan Volga region, such as the Astrakhan Kremlin, the Chetyryohbugorinskoye fortress, Tsaritsyn, Chyorny Yar and Krasny Yar fortresses are given. The author describes the results of studies of the Selitryany town fortress– the only military-industrial facility of the XVIII century, subjected to archaeological study

Counting MSTD Sets in Finite Abelian Groups

In an abelian group G, a more sums than differences (MSTD) set is a subset A of G such that |A+A|>|A-A|. We provide asymptotics for the number of MSTD sets in finite abelian groups, extending previous results of Nathanson. The proof contains an application of a recently resolved conjecture of Alon and Kahn on the number of independent sets in a regular graph.Comment: 17 page

Constructing MSTD Sets Using Bidirectional Ballot Sequences

A more sums than differences (MSTD) set is a finite subset S of the integers such that |S+S| > |S-S|. We construct a new dense family of MSTD subsets of {0, 1, 2, ..., n-1}. Our construction gives Theta(2^n/n) MSTD sets, improving the previous best construction with Omega(2^n/n^4) MSTD sets by Miller, Orosz, and Scheinerman.Comment: 9 pages, 2 tables, 5 figure

The Complex of Mausoleums at the Village of Lapas: on the question of site attribution

The purpose of the study: The paper presents the results of comprehensive archaeological studies on the largest necropolis of the Golden Horde elite in the Ulus of Jochi – Lapas complex of mausoleums. A critical analysis of the information from written sources, materials, obtained during the studies in 1995-1997 and archaeological works on the Lapas complex in 2018-2022, were carried out. Research materials: The materials of topographic works carried out by the Volga Archaeological Expedition IA RAS in 1995-1997 and the data of decryption of remote sensing of the Earth, got by specialists of the Progress Rocket Space Centre in 2013, were used. The results of magnetometric and geodetic studies, conducted at the Lapas complex of mausoleums in 2018-2022, are presented. The results of the first archaeological excavation, conducted in order to study the stratigraphy and planography of the buried site (mausoleum No. 1), are shown. The analysis of the numismatic collection from the site is presented. Results and scientific novelty: Interdisciplinary research, held on the Lapas mausoleum complex, including topographic, geophysical, geodetic and archaeological works, allowed to make a number of assumptions and conclusions about the time of creation, activity and destruction of the complex, its general structure and design features, the potential of the archaeological site, its role and significance in the culture of the Golden Horde. The presented scientific research is pioneering, as earlier comprehensive studies of such level on archaeological sites of the Golden Horde period were not conducted

Assessment of a single trial impact on the amplitude of the averaged event related potentials

Widely used in neuroscience the averaging of event related potentials is based on the assumption that small responses to the investigated events are present in every trial but can be hidden under the random noise. This situation often takes place, especially in experiments performed at hierarchically lower levels of sensory systems. However, in the studies of higher order complex neuronal networks evoked responses might appear only under particular conditions and be absent otherwise. We encountered this problem studying a propagation of interoceptive information to the cortical areas in the sleep-wake cycle. Cortical responses to various visceral events were present during some periods of sleep, then disappeared for a while and restored again after a period of absence. Further investigation of the viscero-cortical communication required a method that would allow labeling the trials contributing to the averaged event related responses–“efficient trials,” and separating them from the trials without any response. Here we describe a heuristic approach to solving this problem in the context of viscero-cortical interactions occurring during sleep. However, we think that the proposed technique can be applicable to any situation where neuronal processing of the same events is expected to be variable due to internal or external factors modulating neuronal activity. The method was first implemented as a script for Spike 2 program version 6.16 (CED). However, at present a functionally equivalent version of this algorithm is also available as Matlab code at https://github.com/george-fedorov/erp-correlations

Coupled Pendulums System Under Control by Vertical Oscillations

В статье рассматривается математическая модель системы, состоящей из двух обратных маятников с упругой связью (пружиной). Система управляется программно, посредством вертикальных осцилляций точки крепления одного из маятников. Проведено исследование динамики указанной механической системы, сформулированы условия, обеспечивающие ее стабилизацию. Построены зоны устойчивости в пространстве исходных параметров. Представлена эволюция зон устойчивости в зависимости от значений жесткости пружины. В работе также приведены результаты численных экспериментов, иллюстрирующих динамику системыIn this paper we propose a mathematical model consisting of two inverse pendulums with an elastic coupling (by spring). We propose a dynamic programmed control of the model motion, implemented through vertical oscillations of the common pendulums pivot point. We investigate dynamics of this mechanical system, and formulated a condition for identifying stability of the system. We constructed stability zones in the spaces of the original and dimensionless parameters. Also, we obtain evolution of stability zones depending on spring stiffness values. In conclusion, we presented results of numerical software experiments for various system configuration

A Neurophysiologically Plausible Population Code Model for Feature Integration Explains Visual Crowding

An object in the peripheral visual field is more difficult to recognize when surrounded by other objects. This phenomenon is called “crowding”. Crowding places a fundamental constraint on human vision that limits performance on numerous tasks. It has been suggested that crowding results from spatial feature integration necessary for object recognition. However, in the absence of convincing models, this theory has remained controversial. Here, we present a quantitative and physiologically plausible model for spatial integration of orientation signals, based on the principles of population coding. Using simulations, we demonstrate that this model coherently accounts for fundamental properties of crowding, including critical spacing, “compulsory averaging”, and a foveal-peripheral anisotropy. Moreover, we show that the model predicts increased responses to correlated visual stimuli. Altogether, these results suggest that crowding has little immediate bearing on object recognition but is a by-product of a general, elementary integration mechanism in early vision aimed at improving signal quality

A Functional Architecture of Optic Flow in the Inferior Parietal Lobule of the Behaving Monkey

The representation of navigational optic flow across the inferior parietal lobule was assessed using optical imaging of intrinsic signals in behaving monkeys. The exposed cortex, corresponding to the dorsal-most portion of areas 7a and dorsal prelunate (DP), was imaged in two hemispheres of two rhesus monkeys. The monkeys actively attended to changes in motion stimuli while fixating. Radial expansion and contraction, and rotation clockwise and counter-clockwise optic flow stimuli were presented concentric to the fixation point at two angles of gaze to assess the interrelationship between the eye position and optic flow signal. The cortical response depended upon the type of flow and was modulated by eye position. The optic flow selectivity was embedded in a patchy architecture within the gain field architecture. All four optic flow stimuli tested were represented in areas 7a and DP. The location of the patches varied across days. However the spatial periodicity of the patches remained constant across days at ∼950 and 1100 µm for the two animals examined. These optical recordings agree with previous electrophysiological studies of area 7a, and provide new evidence for flow selectivity in DP and a fine scale description of its cortical topography. That the functional architectures for optic flow can change over time was unexpected. These and earlier results also from inferior parietal lobule support the inclusion of both static and dynamic functional architectures that define association cortical areas and ultimately support complex cognitive function