Improved Bounds on Fourier Entropy and Min-entropy

Given a Boolean function f:{ -1,1} ^{n}→ { -1,1, define the Fourier distribution to be the distribution on subsets of [n], where each S ⊆ [n] is sampled with probability f ˆ (S) 2 . The Fourier Entropy-influence (FEI) conjecture of Friedgut and Kalai [28] seeks to relate two fundamental measures associated with the Fourier distribution: does there exist a universal constant C > 0 such that H(f ˆ2 ) ≤ C ⋅ Inf (f), where H (fˆ2) is the Shannon entropy of the Fourier distribution of f and Inf(f) is the total influence of f In this article, we present three new contributions toward the FEI conjecture: (1) Our first contribution shows that H(f ˆ2 ) ≤ 2 ⋅ aUC ⊕ (f), where aUC ⊕ (f) is the average unambiguous parity-certificate complexity of f . This improves upon several bounds shown by Chakraborty et al. [20]. We further improve this bound for unambiguous DNFs. We also discuss how our work makes Mansour's conjecture for DNFs a natural next step toward resolution of the FEI conjecture. (2) We next consider the weaker Fourier Min-entropy-influence (FMEI) conjecture posed by O'Donnell and others [50, 53], which asks if H ∞ fˆ2) ≤ C ⋅ Inf(f), where H ∞ fˆ2) is the min-entropy of the Fourier distribution. We show H ∞ (fˆ2) ≤ 2⋅C min ⊕ (f), where C min ⊕ (f) is the minimum parity-certificate complexity of f . We also show that for all ε≥0, we have H ∞ (fˆ2) ≤2 log⁡(∥f ˆ ∥1,ε/(1−ε)), where ∥f ˆ ∥1,ε is the approximate spectral norm of f . As a corollary, we verify the FMEI conjecture for the class of read- k DNFs (for constant k ). (3) Our third contribution is to better understand implications of the FEI conjecture for the structure of polynomials that 1/3-approximate a Boolean function on the Boolean cube. We pose a conjecture: no flat polynomial (whose non-zero Fourier coefficients have the same magnitude) of degree d and sparsity 2 ω(d) can 1/3-approximate a Boolean function. This conjecture is known to be true assuming FEI, and we prove the conjecture unconditionally (i.e., without assuming the FEI conjecture) for a class of polynomials. We discuss an intriguing connection between our conjecture and the constant for the Bohnenblust-Hille inequality, which has been extensively studied in functional analysis.

Comparison of Modeling SPARC spiral galaxies’ rotation curves: halo models vs. MOND

The tension between luminous matter and dynamical matter has long been an interesting and controversial topic in the investigation of galaxies. This is particularly true when we study spiral galaxies for which we have high quality observations of rotation curves. The solutions to the tension are proposed in two different approaches, one is the dark matter hypothesis and the other is MOdified Newtonian Dynamics (MOND) theory. When we test the solutions by using observational data of rotation curves, the controversy arises when we apply them to both low surface brightness (LSB) galaxies and high surface brightness (HSB) galaxies. Usually one likes to use the rotation curves of LSB galaxies, since dark matter is needed or the Newtonian acceleration falls below the characteristic acceleration a 0 in most regions of such galaxies, even near their centers. But for HSB galaxies, dark matter is needed or Newtonian acceleration falls below the characteristic acceleration a 0 only in their outer regions so it is helpful to single out HSB galaxies from some large sample to test the solutions. To this end, we employ a sub-sample of the rotation curves consisting of 45 non-bulgy HSB galaxies selected from the Spitzer Photometry and Accurate Rotation Curves (SPARC) database to test two dark halo models (NFW and Burkert) and MOND. We find that, among the three models, the core-dominated Burkert halo model ( χ ν 2 = 1.00 ) provides a better description of the observed data than the NFW model ( χ ν 2 = 1.44 ) or MOND model ( χ ν 2 = 1.87 ). This is not consistent with the most recent numerical simulations, which tend to favor some cuspy density profiles for HSB galaxies. For MOND, when we take a 0 as a free parameter, there is no obvious correlation between a 0 and disk central surface brightness at 3.6 μm of these HSB spiral galaxies, which is in line with the basic assumption of MOND that a 0 should be a universal constant, but is surprisingly not consistent with the results when LSB galaxies are included. Furthermore, our fittings give a 0 an average value of (0.74 ±0.45) ×10−8 cm s−2, which only marginally supports the standard value of a 0 (1.21 ×10−8 cm s−2). Since the standard value of a 0 is strongly supported when both HSB and LSB galaxies are included in the large SPARC sample, we conclude that our slightly smaller value of a 0 cannot be explained by the so called external field effect in MOND theory.

Free data at spacelike $${\mathscr {I}}$$ and characterization of Kerr-de Sitter in all dimensions

We study the free data in the Fefferman–Graham expansion of asymptotically Einstein $$(n+1)$$ ( n + 1 ) -dimensional metrics with non-zero cosmological constant. We analyze the relation between the electric part of the rescaled Weyl tensor at $${\mathscr {I}}$$ I , D, and the free data at $${\mathscr {I}}$$ I , namely a certain traceless and transverse part of the n-th order coefficient of the expansion $$\mathring{g}_{(n)}$$ g ˚ ( n ) . In the case $$\Lambda <0$$ Λ < 0 and Lorentzian signature, it was known [23] that conformal flatness at $${\mathscr {I}}$$ I is sufficient for D and $$\mathring{g}_{(n)}$$ g ˚ ( n ) to agree up to a universal constant. We recover and extend this result to general signature and any sign of non-zero $$\Lambda $$ Λ . We then explore whether conformal flatness of $${\mathscr {I}}$$ I is also neceesary and link this to the validity of long-standing open conjecture that no non-trivial purely magnetic $$\Lambda $$ Λ -vacuum spacetimes exist. In the case of $${\mathscr {I}}$$ I non-conformally flat we determine a quantity constructed from an auxiliary metric which can be used to retrieve $$\mathring{g}_{(n)}$$ g ˚ ( n ) from the (now singular) electric part of the Weyl tensor. We then concentrate in the $$\Lambda >0$$ Λ > 0 case where the Cauchy problem at $${\mathscr {I}}$$ I of the Einstein vacuum field equations is known to be well-posed when the data at $${\mathscr {I}}$$ I are analytic or when the spacetime has even dimension. We establish a necessary and sufficient condition for analytic data at $${\mathscr {I}}$$ I to generate spacetimes with symmetries in all dimensions. These results are used to find a geometric characterization of the Kerr-de Sitter metrics in all dimensions in terms of its geometric data at null infinity.

The jurisprudence of universal subjectivity: COVID-19, vulnerability and housing

Drawing upon Martha Fineman’s vulnerability theory, the paper argues that the legal claims of homeless appellants before and during the COVID-19 pandemic illustrate our universal vulnerability which stems from the essential, life-sustaining activities flowing from the ontological status of the human body. By recognizing that housing availability has constitutional significance because it provides for life-sustaining activities such as sleeping, eating and lying down, I argue that the legal rationale reviewed in the paper underscores the empirical, ontological reality of the body as the basis for a jurisprudence of universal vulnerability. By tracing the constitutional basis of this jurisprudence from Right to Travel to Eighth Amendment grounds during COVID-19, the paper outlines a distinct legal paradigm for understanding vulnerability in its universal, constant and essential form – one of the central premises of vulnerability theory.

The Mystery of the Lorentz Transform: A Reconstruction and Its Implications for Einstein’s Theories of Relativity and cosmology.

The Lorentz Transformation (LT) is an arbitrary and poorly conceived mathematical tool designed to make Maxwell’s electromagnetism conform to Galilean relativity, which formed the basis of classical mechanics and physics. A strange combination of this transform with an axiomatic assumption by Albert Einstein that the velocity of light c is an absolute and universal constant has led to an idealist, geometrical and phenomenological view of the universe, that is at variance with objective reality. This conundrum that has lasted for more than hundred years has led to rampant mysticism and has impaired the development of positive knowledge of the universe. The present reconstruction of LT shows that the gamma term, which fueled mysticism in physics and cosmology is, on the contrary, a natural outcome of the subjective geometrical rendition of the speed of light and the idealist unification of abstract space and time into a 4D “spacetime” manifold; by Minkowski and Einstein. Only a materialist dialectical perspective of space and time can rid physics of all mysticism arising out of LT; from the quantum to the cosmic.

Financial Return Distributions: Past, Present, and COVID-19

We analyze the price return distributions of currency exchange rates, cryptocurrencies, and contracts for differences (CFDs) representing stock indices, stock shares, and commodities. Based on recent data from the years 2017–2020, we model tails of the return distributions at different time scales by using power-law, stretched exponential, and q-Gaussian functions. We focus on the fitted function parameters and how they change over the years by comparing our results with those from earlier studies and find that, on the time horizons of up to a few minutes, the so-called “inverse-cubic power-law” still constitutes an appropriate global reference. However, we no longer observe the hypothesized universal constant acceleration of the market time flow that was manifested before in an ever faster convergence of empirical return distributions towards the normal distribution. Our results do not exclude such a scenario but, rather, suggest that some other short-term processes related to a current market situation alter market dynamics and may mask this scenario. Real market dynamics is associated with a continuous alternation of different regimes with different statistical properties. An example is the COVID-19 pandemic outburst, which had an enormous yet short-time impact on financial markets. We also point out that two factors—speed of the market time flow and the asset cross-correlation magnitude—while related (the larger the speed, the larger the cross-correlations on a given time scale), act in opposite directions with regard to the return distribution tails, which can affect the expected distribution convergence to the normal distribution.

LEXICAL AND WORLD-MODELING ACTIVITY OF SOMATISMS AND ANTHROPOMORPHISM OF LANGUAGE CATEGORIZATION IN THE PROVERB

Введение. Рассматривается лексическая и миромоделирующая активность единиц лексико-семантической группы «Части тела» ‒ соматизмов, находящая отражение в текстах русских народных пословиц. Особенности семантики и прагматики соматизмов, обусловливающие специфику их функционирования в фольклорном тексте, позволяют определять соматическую лексику в качестве маркеров национальной идентичности. Целью исследования является изучение соматизмов, функционирующих в текстах русских народных пословиц, в аспекте реализации ими своего лексического и миромоделирующего потенциала. Материал и методы. В качестве материала исследования привлекаются тексты русских народных пословиц, содержащих лексемы-соматизмы. Принцип отбора эмпирического материала ‒ на основании сплошной выборки наиболее частотно встречающихся соматических единиц из текстов. Методологию исследования составляют методы наблюдения, количественного анализа, лексико-семантического анализа с привлечением элементов дискурсивного и концептуального анализа. Результаты и обсуждение. Соматизмы, значение которых строится на основе смыслов антропоморфности, играют значительную роль в формировании представления о человеке в языковой и концептуальной картине мира. Концептуальный смысл соматизмов проявляется неодинаково в разных лингвокультурах. При наличии универсальных, константных характеристик, свойственных всем этносам, наблюдается присутствие трактовок, обусловленных спецификой той или иной культуры. Это становится очевидным при сопоставлении случаев функционирования соматизмов в текстах русских и китайских пословиц: названные лингвокультуры чрезвычайно различаются в культурном и языковом планах. Выявлено, что наибольшим лексическим и миромоделирующим потенциалом, судя по текстам пословиц, в русской языковой картине мира обладают соматизмы голова, рука, глаза. За каждой соматической лексемой закреплен конкретный концептуальный смысл, важной составляющей частью которого является аксиологический компонент «ценность». Так, соматизм голова интерпретируется как «ценность интеллекта», рука ‒ «ценность жизненной активности», глаза ‒ «ценность личного участия». В меньшем количестве в пословицах присутствуют соматизмы волосы, ноги, рот, язык, нос. В этом перечне в первую очередь очевидны такие интерпретации, как ноги, символизирующие «ценность мобильности», и волосы ‒ маркер антиценности «внешнего» в противовес ценности «внутреннего». Заключение. Изучение соматизмов в аспекте рассмотрения их лексической и миромоделирующей активности, проявляющейся в фольклорных текстах (в данном случае в пословицах), позволяет формировать представление о фрагментах языковой и концептуальной картины мира этноса. Introduction. The article is devoted to the consideration of the lexical and world-modeling activity of units of the lexical-semantic group «Parts of the body» - somatisms, which is reflected in the texts of Russian folk proverbs. The peculiarities of the semantics and pragmatics of somatisms, which determine the specifics of their functioning in a folklore text, make it possible to define somatic vocabulary as markers of national identity. Aim and objectives. The aim of the research is to study the somatisms that function in the texts of Russian folk proverbs, in the aspect of their realization of their lexical and world-modeling potential. Material and methods. As the research material, the texts of Russian folk proverbs containing somatism lexemes are used. The principle of selection of empirical material is based on a continuous sample of the most frequently encountered somatic units from texts. The research methodology consists of methods of observation, quantitative analysis, lexical and semantic analysis, with the involvement of elements of discourse and conceptual analysis. Results and discussion. Somatisms, the meaning of which is based on the meanings of anthropomorphism, play a significant role in the formation of the idea of a person in the linguistic and conceptual picture of the world. The conceptual meaning of somatisms is manifested differently in different linguocultures. In the presence of undoubted universal, constant characteristics inherent in all ethnic groups, there is a presence of interpretations due to the specificity of a particular culture. This becomes obvious when comparing the cases of the functioning of somatisms in the texts of Russian and Chinese proverbs: the named linguocultures are extremely different in cultural and linguistic terms. It was revealed that the greatest lexical and world-modeling potential, judging by the texts of proverbs, in the Russian linguistic picture of the world is possessed by the somatisms head, hand, and eyes. Each somatic lexeme has a specific conceptual meaning, an important component of which is the axiological component “value”. So, somatism, the head is interpreted as «the value of the intellect», the hand is the «value of vital activity», the eyes are the «value of personal participation.» In fewer proverbs, there are somatisms hair, legs, mouth, tongue, nose. In this list, interpretations such as legs, symbolizing the «value of mobility,» and hair, a marker of the anti-value of «external» as opposed to the value of «internal», are primarily evident. Conclusion. The study of somatisms in the aspect of considering their lexical and world-modeling activity, manifested in folklore texts (in this case, in proverbs), makes it possible to form an idea of fragments of the linguistic and conceptual picture of the world of an ethnic group.

Introduction to The Origin of Electromagnetic Mass (Electromagnetic Inertia)

Newton described in his second law of motion the classical definition of mass (inertia). However, it is impossible to calculate with Newton’s second law of motion the (electromagnetic) mass of a beam of light (Ref. [1], [2],[3]). Because the speed of light is a universal constant which follows from Albert Einstein’s Theory of Special Relativity, it is impossible to accelerate or to slow down a beam of light and for that reason it is impossible to determine the electromagnetic mass of a beam of light (free electromagnetic radiation) by Newton’s second law. To calculate the electromagnetic mass of free or confined electromagnetic radiation, the fundamental concept of the New Theory has been used that the Universe is in a perfect Equilibrium and that any electromagnetic field configuration is in a perfect equilibrium with itself and its surrounding. From this fundamental concept follows a different definition of (confined) electromagnetic mass. Electromagnetic mass (or inertia) has been determined by the relativistic Lorentz transformation of the radiation pressures in all different directions and the disturbance of a uniform motion (or position at rest) of confined electromagnetic radiation results in a relativistic effect which we measure (experience) as electromagnetic mass (inertia). The mass in [kg] of an object will be generally measured by acceleration (or deceleration) of the object according Newton’s second law of motion. In the theory of special relativity, the speed of light is a fundamental constant and the intensity of the light is not a universal constant. The calculate the relativistic mass of Confined Electromagnetic Radiation, we start with a thought experiment in which a beam of light is propagating between two 100 % reflecting mirrors, indicated as Mirror A and Mirror B. Both mirrors are part of a rigid construction and the relative velocity between both mirrors always equals zero. The results of this calculation will be generalized for any kind of electromagnetic radiation which has been confined by its own electromagnetic and gravitational field. When the speed of the observer has the same speed as the speed of the light source, then the observer and the light source are relative at rest. And the same light intensity will be measured at the location of the emitter and at the location of the observer.

The Origin of Electromagnetic Mass (Electromagnetic Inertia)

Newton described in his second law of motion the classical definition of mass (inertia). However, it is impossible to calculate with Newton’s second law of motion the (electromagnetic) mass of a beam of light. Because the speed of light is a universal constant which follows from Albert Einstein’s Theory of Special Relativity, it is impossible to accelerate or to slow down a beam of light and for that reason it is impossible to determine the electromagnetic mass of a beam of light (free electromagnetic radiation) by Newton’s second law. To calculate the electromagnetic mass of free or confined electromagnetic radiation, the fundamental concept of the New Theory has been used that the Universe is in a perfect Equilibrium and that any electromagnetic field configuration is in a perfect equilibrium with itself and its surrounding. From this fundamental concept follows a different definition of (confined) electromagnetic mass. Electromagnetic mass (or inertia) has been determined by the relativistic Lorentz transformation of the radiation pressures in all different directions and the disturbance of a uniform motion (or position at rest) of confined electromagnetic radiation results in a relativistic effect which we measure (experience) as electromagnetic mass (inertia). The mass in [kg] of an object will be generally measured by acceleration (or deceleration) of the object according Newton’s second law of motion. In the theory of special relativity, the speed of light is a fundamental constant and the intensity of the light is not a universal constant. The calculate the relativistic mass of Confined Electromagnetic Radiation, we start with a thought experiment in which a beam of light is propagating between two 100 % reflecting mirrors, indicated as Mirror A and Mirror B. Both mirrors are part of a rigid construction and the relative velocity between both mirrors always equals zero. The results of this calculation will be be generalized for any kind of electromagnetic radiation which has been confined by its own electromagnetic and gravitational field. When the speed of the observer has the same speed as the speed of the light source, then the observer and the light source are relative at rest. And the same light intensity will be measured at the location of the emitter and at the location of the observer.