More on discrete convexity

In several recent papers some concepts of convex analysis were extended to discrete sets. This paper is one more step in this direction. It is well known that a local minimum of a convex function is always its global minimum. We study some discrete objects that share this property and provide several examples of convex families related to graphs and to two-person games in normal form

On Nash-Solvability of Finite Two-Person Tight Vector Game Forms

We consider finite two-person normal form games. The following four properties of their game forms are equivalent: (i) Nash-solvability, (ii) zero-sum-solvability, (iii) win-lose-solvability, and (iv) tightness. For (ii, iii, iv) this was shown by Edmonds and Fulkerson in 1970. Then, in 1975, (i) was added to this list and it was also shown that these results cannot be generalized for $n$-person case with $n > 2$. In 1990, tightness was extended to vector game forms ($v$-forms) and it was shown that such $v$-tightness and zero-sum-solvability are still equivalent, yet, do not imply Nash-solvability. These results are applicable to several classes of stochastic games with perfect information. Here we suggest one more extension of tightness introducing $v^+$-tight vector game forms ($v^+$-forms). We show that such $v^+$-tightness and Nash-solvability are equivalent in case of weakly rectangular game forms and positive cost functions. This result allows us to reduce the so-called bi-shortest path conjecture to $v^+$-tightness of $v^+$-forms. However, both (equivalent) statements remain open

Polynomial algorithms computing two lexicographically safe Nash equilibria in finite two-person games with tight game forms given by oracles

In 1975 the first author proved that every finite tight two-person game form $g$ is Nash-solvable, that is, for every payoffs $u$ and $w$ of two players the obtained game $(g;u,w)$, in normal form, has a Nash equilibrium (NE) in pure strategies. This result was extended in several directions; here we strengthen it further. We construct two special NE realized by a lexicographically safe (lexsafe) strategy of one player and a best response of the other. We obtain a polynomial algorithm computing these lexsafe NE. This is trivial when game form $g$ is given explicitly. Yet, in applications $g$ is frequently realized by an oracle \cO such that size of $g$ is exponential in size |\cO| of \cO. We assume that game form g = g(\cO) generated by \cO is tight and that an arbitrary {\em win-lose game} $(g;u,w)$ (in which payoffs $u$ and $w$ are zero-sum and take only values $\pm 1$) can be solved, in time polynomial in |\cO|. These assumptions allow us to construct an algorithm computing two (one for each player) lexsafe NE in time polynomial in |\cO|. We consider four types of oracles known in the literature and show that all four satisfy the above assumptions

On Nash-solvability of n-person graphical games under Markov's and a priori realizations

We consider graphical $n$-person games with perfect information that have no Nash equilibria in pure stationary strategies. Solving these games in mixed strategies, we introduce probabilistic distributions in all non-terminal positions. The corresponding plays can be analyzed under two different basic assumptions: Markov's and a priori realizations. The former one guarantees existence of a uniformly best response of each player in every situation. Nevertheless, Nash equilibrium may fail to exist even in mixed strategies. The classical Nash theorem is not applicable, since Markov's realizations may result in the limit distributions and effective payoff functions that are not continuous. The a priori realization does not share many nice properties of the Markov one (for example, existence of the uniformly best response) but in return, Nash's theorem is applicable. We illustrate both realizations in details by two examples with $2$ and $3$ players and also provide some general results

Logical Contradictions in the One-Way ANOVA and Tukey–Kramer Multiple Comparisons Tests with More Than Two Groups of Observations

We show that the one-way ANOVA and Tukey–Kramer (TK) tests agree on any sample with two groups. This result is based on a simple identity connecting the Fisher–Snedecor and studentized probabilistic distributions and is proven without any additional assumptions; in particular, the standard ANOVA assumptions (independence, normality, and homoscedasticity (INAH)) are not needed. In contrast, it is known that for a sample with k>2 groups of observations, even under the INAH assumptions, with the same significance level α, the above two tests may give opposite results: (i) ANOVA rejects its null hypothesis H0A:μ1=…=μk, while the TK one, H0TK(i,j):μi=μj, is not rejected for any pair i,j∈{1,…,k}; (ii) the TK test rejects H0TK(i,j) for a pair (i,j) (with i≠j), while ANOVA does not reject H0A. We construct two large infinite pseudo-random families of samples of both types satisfying INAH: in case (i) for any k≥3 and in case (ii) for some larger k. Furthermore, case (ii) ANOVA, being restricted to the pair of groups (i,j), may reject equality μi=μj with the same α. This is an obvious contradiction, since μ1=…=μk implies μi=μj for all i,j∈{1,…,k}. Such contradictions appear already in the symmetric case for k=3, or in other words, for three groups of d,d, and c observations with sample means +1,−1, and 0, respectively. We outline conditions necessary and sufficient for this phenomenon. Similar contradictory examples are constructed for the multivariable linear regression (MLR). However, for these constructions, it seems difficult to verify the Gauss–Markov assumptions, which are standardly required for MLR. Mathematics Subject Classification: 62 Statistics

Lexicographically maximal edges of dual hypergraphs and Nash-solvability of tight game forms

Let $\mathcal{A} = \{A_1, \ldots, A_m\}$ and $\mathcal{B} = \{B_1, \ldots, B_n\}$ be a pair of dual multi-hypergraphs on the common ground set $O = \{o_1, \ldots, o_k\}$. Note that each of them may have embedded or equal edges. An edge is called containment minimal (or just minimal, for short) if it is not a strict superset of another edge. Yet, equal minimal edges may exist. By duality, (i) $A \cap B \neq \emptyset$ for every pair $A \in \mathcal{A}$ and $B \in \mathcal{B}$; (ii) if $A$ is minimal then for every $o \in A$ there exists a $B \in \mathcal{B}$ such that $A \cap B = \{o\}$. We will extend claim (ii) as follows. A linear order $\succ$ over $O$ defines a unique lexicographic order $\succ_L$ over the $2^O$. Let $A$ be a lexicographically maximal (lexmax) edge of $\mathcal{A}$. Then, (iii) $A$ is minimal and for every $o \in A$ there exists a minimal $B \in \mathcal{B}$ such that $A \cap B = \{o\}$ and $o \succeq o'$ for each $o' \in B$. This property has important applications in game theory implying Nash-solvability of tight game forms as shown in the old (1975 and 1989) work of the first author. Here we give a new, very short, proof of (iii). Edges $A$ and $B$ mentioned in (iii) can be found out in polynomial time. This is trivial if $\mathcal{A}$ and $\mathcal{B}$ are given explicitly. Yet, it is true even if only $\mathcal{A}$ is given, and not explicitly, but by a polynomial containment oracle, which for a subset $O_A \subseteq O$ answers in polynomial time whether $O_A$ contains an edge of $\mathcal{A}$

Содержание нейроспецифических пептидов, маркеров нейромессенджера и нейрорецептора в сыворотке крови детей с вариативными сенсорными расстройствами, легкими когнитивными нарушениями и другой нейропатологией

Background. The role of recently discovered neurospecific peptides in the pathogenesis of acute and progressive neurologic disorders, their neuroprotective features, and possibilities to use them as markers for the course and prognosis of certain diseases have been actively studied in recent decades. However, neurospecific peptides are almost not studied in chronic residual diseases. In our study we measured the levels of neurospecific peptides and some other markers to achieve understanding of general neurophysiological trends in congenital and acquired chronic non-progressive brain pathology with reference to the selection of relevant groups — study objects. Objective. The aim of the study is to study patterns of neurospecific peptides, neurotransmitters and neuroreceptor markers distribution in the serum of children with various pathogenetic variants of chronic neuropathology. Methods. The study included children from 3 to 16 years old with different pathologies. The sample was divided into groups by pathology type: no sensory and neurological disorders, congenital sensory deficit due to mutation of genes expressed and not expressed in the brain, early acquired sensory deficit of multifactorial nature, congenital mild and severe organic disorders of central nervous system (CNS) in residual stage without baseline sensory deficit, acquired functional CNS disorders without baseline organic defect and sensory deficit. The following laboratory data (neurophysiological components) was studied: nerve growth factor, brain-derived neurotropic factor, neurotrophin-3, neurotrophin-4, neuregulin-1-beta-1, beta-secretase, sirtuin-1, synaptophysin, neuronal nitric oxide synthase, and anti-NR2 glutamate receptor antibodies. The parameters of cognitive activity, sense of vision, sense of smell, and acoustic sense were also evaluated. Results. The study included 274 participants. Neuropeptides and markers have shown a variable degree and range in the group spectrum of differences from normal levels. The most variable in the examined sample was NO-synthase, as well as levels of both neurotrophins, beta-secretase, and glutamate receptor marker. All visual deficits were associated with increased NO-synthase levels (p < 0.001). Neuroplasticity peptides (beta-secretase, neurotrophin-3 and 4) have been activated in all pathological conditions. Nerve growth factor and brain-derived neurotropic factor were specifically activated in mild organic CNS lesions (mild cognitive impairments), while neuregulin — in congenital genetically determined visual deficits. There was no specific activation of neuropeptides and NO-synthase level tended to decrease in cases of severe CNS lesions. Conclusion. The study results suggest that all types of early visual impairment are associated with increased physiological neuronal activity, and non-organic neurological functional disorders — mainly with increased physiological synaptic activity. General neuroplasticity processes were activated in all cases of visual deficits but more specific. However, more specific and well-studied processes were activated in mild organic CNS lesions, and neuroplasticity processes did not activate adequately in severe organic CNS lesions probably due to the limited neuronal and synaptic resources.Обоснование. В последние десятилетия активно исследуются вклад недавно открытых нейроспецифических пептидов в патогенез ряда острых и прогрессирующих заболеваний нервной системы, их нейропротективные свойства и возможности их использования для маркирования течения и прогноза некоторых заболеваний. При этом нейроспецифические пептиды почти не изучаются при хронических резидуальных состояниях. В нашем исследовании мы использовали определение уровней нейроспецифических пептидов и некоторых других маркеров для достижения понимания генеральных нейрофизиологических тенденций при врожденной и приобретенной хронической непрогрессирующей патологии мозга на основании подбора соответствующих групп — объектов исследования. Цель исследования — изучить закономерности распределения комплекса нейроспецифических пептидов, маркеров нейромессенджера и нейрорецептора в сыворотке крови детей с различными патогенетическими вариантами хронической нейропатологии. Методы. В исследование были включены дети с различной патологией в возрасте от 3 до 16 лет. Выборка была поделена на группы по типу патологии: отсутствие сенсорных и неврологических нарушений, врожденный сенсорный дефицит вследствие мутации генов, экспрессируемых и не экспрессируемых в мозге, рано приобретенный сенсорный дефицит полиэтиологической природы, врожденные легкие и тяжелые органические нарушения функций центральной нервной системы (ЦНС) в резидуальной стадии без исходного сенсорного дефицита, приобретенные функциональные расстройства ЦНС без исходного органического дефекта и сенсорного дефицита. В батарею измеряемых лабораторно в крови нейрофизиологических компонентов включили фактор роста нервов, нейротрофический фактор мозга, нейротрофин-3, нейротрофин-4, нейрегулин-1-бета-1, бета-секретазу, сиртуин-1, синаптофизин, нейрональную синтазу оксида азота и антитела к глутаматному рецептору NR2. Также оценивались параметры когнитивной деятельности, зрения, обоняния и слухового восприятия. Результаты. В исследование включены 274 участника. Установлено, что нейропептиды и маркеры показали вариативную степень и широту по групповому спектру отличий от нормы. Наиболее изменчивой в обследуемой выборке показала себя NO-синтаза, также часто различались уровни обоих нейротрофинов, бета-секретазы и маркера рецептора глутамата. При любых дефицитах зрения с большой достоверностью был повышен уровень NO-синтазы (р < 0,001). При всех патологических состояниях активировались пептиды нейропластичности — бета-секретаза, нейротрофины-3 и -4. При легких органических поражениях ЦНС (легкие когнитивные нарушения) специфично активировались фактор роста нервов и мозговой нейротрофический фактор, а при врожденных генетически детерминированных зрительных дефицитах специфично активировался нейрегулин. При тяжелых поражениях ЦНС специфической активации нейропептидов не определялось, а уровень NO-синтазы демонстрировал тенденцию к снижению по сравнению с нормой. Заключение. Результаты исследования позволяют предположить, что при всех типах раннего слабовидения происходит повышенная напряженность физиологической нейрональной деятельности, а при неорганических неврологических функциональных расстройствах — преимущественно повышение физиологической синаптической активности. При всех дефицитах зрения активированы процессы общей нейропластичности, но более специфические и изученные из них активируются при легких органических поражениях ЦНС, при тяжелых же органических поражениях ЦНС процессы нейропластичности недостаточно активны, вероятно, вследствие ограниченности нейрональных и синаптических ресурсов