Calculus of Variations

We improve some recent results of Sagiv and Steinerberger that quantify the following uncertainty principle: for a function f with mean zero, either the size of the zero set of the function or the cost of transporting the mass of the positive part of f to its negative part must be big. We also provide a sharp upper estimate of the transport cost of the positive part of an eigenfunction of the Laplacian. This proves a conjecture of Steinerberger and provides a lower bound of the size of the nodal set of the eigenfunction.

few decades ago, (1935), there was a general consensus amongst a group of eight** prominent French mathematicians and academicians that there was a serious shortfall in the disparate mathematical accumulata of the past centuries up to the current time. This was based, in part, on the implicit (and unwarranted) assumption, that there must be one basic mathematical discipline, capable of subsuming all the others by being a provider of the `lingua Franca', in which to express and inter-relate all mathematical thoughts, past present and future, with absolute rigour. This `flew in the face' of † Gödel's Incompleteness Theorem, which theorem seemed to say, (when translated into lay terms), that most (or all) mathematical disciplines were incomplete, in that one could not prove absolute consistency, (and therefore completeness), with the particular language of that discipline for that very same branch of mathematics, but had to resort to the language of another branch of mathematics to try to do so. These eight men named themselves with the collective pseudonym "Nicholas Bourbaki", in which personification they published their findings. They selectively absorbed others into this elite group with the only preconditions being acceptance by other members and that the maximum age of any member was 50. However both Gödel and the group of Bourbachistes overlooked, (or were oblivious of), their common implicit assumption, namely, that there could ever be a complete translation of any, and every, area of mathematics in terms of solely one single, (and singular), conceptual framework. This great expectation of, (and concomitant search for), an elusive umbral mathematical panacea, as though it did indeed exist, (or could ever exist), is the implicit assumption mentioned above, since no such remedy can ever be evolved by Man for mankind, due to the inherent limitations, (and very construction), of the human brain. For instance, one cannot use arithmetic to prove all geometrical theorems, nor geometry to prove all arithmetical theorems. This example is typical of most realms of theory, ideology, philosophy and theosophy, and hinges on the incontrovertible reality of there being mutually exclusive areas, specific to one discipline, doctrine or set of concepts, but not to another. For thousands of years mathematicians, (both amateur and professional), sought in vain to `square' the circle. We now know that their time was wasted, since a rigorous proof was constructed in the 20 th century, that proves the paradoxical nature of that set task. Note well that prior to this proof the task was no less impossible, and there must have been a strong suspicion that it was futile and that sooner or later such a proof of impossibility would, (hopefully), come to light. An analogous situation may pertain in the realm of formalising mathematics:-1) as attempted by Bourbaki in a (mostly) successful series of publications over at least three decades, 2) as refuted and discredited by Gödel with his Theory of Incompleteness and 3) as still being attempted currently by transcendental methods in abstract mathematics. If we take Gödel seriously, we should consider whether this (contemporary and simultaneous) attempted unification of all mathematical ideas in one metalanguage is `flying in the face of logic' and no better (in effect) than trying to square the circle. We, who cannot construct such a proof ourselves, are merely awaiting the proof of impossibility (hopefully) yet to come, as did the would-be `circle squarers'.

It is unlikely that the readers of the original translation of the Rubaiyat of Omar Khayyam are aware of Gelett Burgess, let alone his satirical `take' of that famous poem. The original was written in Persian and there have been other translations, but a parody/transmogrification/satire is rare indeed. The Persian quatrains mis-attributed to Omar Khayyam (1048Khayyam ( -1131) ) number between 1,200-2,000 and were written by many different Persians through the ages, hence the well-known 100-121 English quatrains comprise many of doubtful authenticity. They have been translated into dozens of different languages, of which the English ones were by many different Persian/Iranian scholars. The most notable English translations were by Edward Fitzgerald in 1859 and 1889. By his own admission, they were transmogrified in their translation. To a large extent they can be considered to be original poetry by Fitzgerald, loosely based upon Omar's quatrains, rather than `translated' in the narrow sense. Another English translator was Edward Heron-Allen in 1898. Omar Khayyam was principally an astronomer and mathematician and was labelled "The Astronomer-Poet of Persia". "The Rubaiyat of Omar Cayenne" by Gelett Burgess (1866-1951) is a satirical peppery poem written in the early 20th century, a period characterised by rapid changes in literature and art. This book serves as a parody of the famous "Rubaiyat of Omar Khayyam" and explores themes such as modern literature, the publishing industry, and the nature of creativity amidst commercialism. With a humorous tone, it critiques contemporary literary trends and the pressures faced by authors. In this playful work, Burgess employs a quatrain format to articulate his observations and frustrations about the state of literature and writing. He addresses the challenges that authors encounter, from the overwhelming number of publications to the fickle tastes of readers and critics.

We say that a metrizable space M is a Krasinkiewicz space if any map from a metrizable compactum X into M can be approximated by Krasinkiewicz maps (a map g : X → M is Krasinkiewicz provided every continuum in X is either contained in a fiber of g or contains a component of a fiber of g). In this paper we establish the following property of Krasinkiewicz spaces: Let f : X → Y be a perfect map between metrizable spaces and M a Krasinkiewicz complete AN R-space. If Y is a countable union of closed finite-dimensional subsets, then the function space C(X, M ) with the source limitation topology contains a dense G δ -subset of maps g such that all restrictions g|f -1 (y), y ∈ Y , are Krasinkiewicz maps. The same conclusion remains true if M is homeomorphic to a closed convex subset of a Banach space and X is a C-space.

We say that a space M is a Krasinkiewicz space if for any compactum X the function space C(X, M) contains a dense subset of Krasinkiewicz maps. Recall that a map g: X → M, where X is compact, is said to be Krasinkiewicz [6] if every continuum in X is either contained in a ...

In this paper we present a new verification theorem for optimal stopping problems for Hunt processes. The approach is based on the Fukushima-Dynkin formula [5] and its advantage is that it allows us to verify that a given function is the value function without using the viscosity solution argument. Our verification theorem works in any dimension. We illustrate our results with some examples of optimal stopping of reflected diffusions and absorbed diffusions.

In this paper we present a new verification theorem for optimal stopping problems for Hunt processes. The approach is based on the Fukushima-Dynkin formula, and its advantage is that it allows us to verify that a given function is the value function without using the viscosity solution argument. Our verification theorem works in any dimension. We illustrate our results with some examples of optimal stopping of reflected diffusions and absorbed diffusions.

In Early Transcendentals (The American Mathematical Monthly, Vol. 104, No 7) Steven Weintraub presents a rigorous justifcation of the "early transcendental" calculus textbook approach to the exponential and logarithmic functions. However, he uses tools such as term-by-term differentiation of infinite series. We present a rigorous treatment of the early transcendental approach suitable for a first course in analysis, using mainly the supremum property of the real numbers.

In this paper we derive the best constant for the following Gagliardo-Nirenberg interpolation inequality \begin{eqnarray*} \|u\|_{L^{m+1}}\leq C_{q,m,p} \|u\|^{1-\theta}_{L^{q+1}}\|\nabla u\|^{\theta}_{L^p},\quad \theta=\frac{pd(m-q)}{(m+1)[d(p-q-1)+p(q+1)]}, \end{eqnarray*} where parameters $q,m,p$ respectively belong to the following two ranges: (i) $p>d\geq 1$, $q\geq0$ and $m=\infty$. That shows $L^{\infty}$-type Gagliardo-Nirenberg interpolation inequality. (ii) $p>\max\{1,\frac{2d}{d+2}\}$, $0\leq q<\sigma-1$, and $q<m<\sigma$, where $\sigma$ is defined by $ \sigma:= \frac{(p-1)d+p }{d-p}$ if $p<d$; $\sigma:=\infty $ if $p\geq d$. That gives $L^{m}$-type Gagliardo-Nirenberg interpolation inequality. The best constant $C_{q,m,p}$ is given by \begin{eqnarray*} C_{q,m,p}:=\theta^{-\frac{\theta}{p}}(1-\theta)^{\frac{\theta}{p}-\frac{1}{m+1}}M_c^{-\frac{\theta}{d}},\quad M_c:=\int_{\mathbb{R}^d}u_{c,m}^{q+1}\,dx, \end{eqnarray*} where $u_{c,m}$ is the unique radia...

In this paper, we study the problem of minimizing the first eigenvalue of the p − p- Laplacian plus a potential with weights when the potential and the weight are allowed to vary in the class of rearrangements of a given fixed potential V 0 V_0 and weight g 0 g_0 . Our results generalize those obtained in earlier papers.

Данная работа будет целиком построена на фундаменте предшествующей ей работы [1] и унаследует ту ее особенность, что будет обеднена исчерпывающей доказательной базой ввиду ограничений формата, не позволяющего включения в текст объемистых выдержек из теорий, по логической структуре своей тотально закольцованных в отношении утверждений любого условно самостоятельного раздела. Также полезно будет предупредить, что если человеку традиционных воззрений некоторые выкладки из работы [1] могут показаться неожиданными, то демонстрируемые ниже связки установлений определенно имеют шанс быть им по той же шкале оценены как надуманные. Автор свой шок при взгляде на выявляемиые цифры переживал призвав на помощь всё отпущенное ему природой чувство юмора, и это вовсе не дерзкий эмоциональный оффтоп противника научной этики, а взвешенное предупреждение. Автор не может предугадать как будут справляться с собственным возмущением люди не приветствующие лабильность воззрений в своем познании мира, в частности те из них, и это банальнейший пример, чье сознание основательнее премлемого уровня критичности очаровано инфляционной мифологемой космологии, но своим опытом купирования переживаний он, вероятнее всего вовремя, счел полезным поделиться. Пример расчета массы бозона Хиггса служит буквально вопиющей иллюстрацией изложенного в [1] представления о пирамидальном усложнении требующегося науке математического инструментария по мере дедуктивного восхождения скрытой логики самого природного устроения от простого к сложному: этот расчет фундаментально примитивен. Англоязычному читателю автор приносит традиционные извинения за стабильный-по меткому выражению Дональда Трампа-перевод с русского. Также следует отметить такую особенность изложения, обусловленную как ограничениями, так и преимуществами метафизического инструментария перед физическим, проясняемыми еще в работе [1], что каждое этапное рассуждение в любом отдельном дедуктивистском построении сопровождается предложением некой неприкрыто искусственной умозрительной визуализации обсуждаемого явления с предъявлением взору наблюдателя некоторых геометризующихся представлений об объекте изучения, которые в физическом мире (6 + 1)-мерного пространства не имеют особого смысла даже в качестве проекций явления в трехили четырехмерность, и поляроидный снимок мгновения, будь он возможен, показал бы фотомастеру иную фигуру. В метафизическом мире предваряющих физику фундаментальных установлений, однако, визуализация процессов играет свою важную объяснительскую, кумулятивно-мнемоническую и генеративную по отношению к новым логическим ходам упрощенческую роль. Метафизика пробивается к результату не столь последовательно и даже не столь научно как то характерно для физики, установившей свои каноны научности, но важен именно результат, легко проверяемый затем идуктивистским экспериментом последней, а не обломки осыпающихся за спиной метафизика наивных или не самых удачных вспомогательных

Данная работа будет целиком построена на фундаменте предшествующей ей работы [1] и унаследует ту ее особенность, что будет обеднена исчерпывающей доказательной базой ввиду ограничений формата, не позволяющего включения в текст объемистых выдержек из теорий, по логической структуре своей тотально закольцованных в отношении утверждений любого условно самостоятельного раздела. Также полезно будет предупредить, что если человеку традиционных воззрений некоторые выкладки из работы [1] могут показаться неожиданными, то демонстрируемые ниже связки установлений определенно имеют шанс быть им по той же шкале оценены как надуманные. Автор свой шок при взгляде на выявляемиые цифры переживал призвав на помощь всё отпущенное ему природой чувство юмора, и это вовсе не дерзкий эмоциональный оффтоп противника научной этики, а взвешенное предупреждение. Автор не может предугадать как будут справляться с собственным возмущением люди не приветствующие лабильность воззрений в своем познании мира, в частности те из них, и это банальнейший пример, чье сознание основательнее премлемого уровня критичности очаровано инфляционной мифологемой космологии, но своим опытом купирования переживаний он, вероятнее всего вовремя, счел полезным поделиться. Пример расчета массы бозона Хиггса служит буквально вопиющей иллюстрацией изложенного в [1] представления о пирамидальном усложнении требующегося науке математического инструментария по мере дедуктивного восхождения скрытой логики самого природного устроения от простого к сложному: этот расчет фундаментально примитивен. Англоязычному читателю автор приносит традиционные извинения за стабильный-по меткому выражению Дональда Трампа-перевод с русского. Также следует отметить такую особенность изложения, обусловленную как ограничениями, так и преимуществами метафизического инструментария перед физическим, проясняемыми еще в работе [1], что каждое этапное рассуждение в любом отдельном дедуктивистском построении сопровождается предложением некой неприкрыто искусственной умозрительной визуализации обсуждаемого явления с предъявлением взору наблюдателя некоторых геометризующихся представлений об объекте изучения, которые в физическом мире (6 + 1)-мерного пространства не имеют особого смысла даже в качестве проекций явления в трехили четырехмерность, и поляроидный снимок мгновения, будь он возможен, показал бы фотомастеру иную фигуру. В метафизическом мире предваряющих физику фундаментальных установлений, однако, визуализация процессов играет свою важную объяснительскую, кумулятивно-мнемоническую и генеративную по отношению к новым логическим ходам упрощенческую роль. Метафизика пробивается к результату не столь последовательно и даже не столь научно как то характерно для физики, установившей свои каноны научности, но важен именно результат, легко проверяемый затем идуктивистским экспериментом последней, а не обломки осыпающихся за спиной метафизика наивных или не самых удачных вспомогательных

Οι συναρτησιακές εξισώσεις αποτελούν μία από τις πιο ενδιαφέρουσες περιοχές των μαθηματικών, καθώς συνδέουν άμεσα την αλγεβρική σκέψη με την αναλυτική μέθοδο. Στο παρόν σύγγραμμα συγκεντρώνονται βασικά θεωρήματα, χαρακτηριστικές κλασικές εξισώσεις (Cauchy, Jensen, d’Alembert), καθώς και μία πλούσια συλλογή λυμένων ασκήσεων σε κλιμακούμενη δυσκολία – από τις απλές έως τις «ολυμπιακού» επιπέδου. Στόχος είναι να παρουσιαστεί όχι μόνο η τεχνική επίλυσης, αλλά και η μεθοδολογία: χρήση ειδικών τιμών, επαγωγή, συμμετρίες (άρτια–περιττά), υπόθεση συνέχειας ή μονοτονίας, και αντιμετώπιση παθολογικών λύσεων. Το βιβλίο μπορεί να λειτουργήσει ως βοήθημα σε μαθητές, φοιτητές, αλλά και σε διδάσκοντες που αναζητούν παραδείγματα για την τάξη ή για μαθηματικούς διαγωνισμούς.

Η παρουσίαση είναι αυτοτελής: κάθε κεφάλαιο ξεκινά με τις κλασικές μορφές και καταλήγει σε πιο σύνθετα προβλήματα, ενώ στο τέλος παρατίθεται γλωσσάριο ορολογίας για διευκόλυνση.
ΓΙΑΝΝΗΣ ΠΛΑΤΑΡΟΣ

This article presents the problem of quickest descent, or the Brachistochrone curve, that may be solved by the calculus of variations and the Euler-Lagrange equation. The cycloid is the quickest curve and also has the property of isochronism by which Huygens improved on Galileo's pendulum.

These are a few excerpts from the book of the same name, which is being prepared for publication in English translation. Based on the analogy between information processes in scientific & information activities and in nature in the context of the Bradford-type distribution, or the well-known Bradford distribution here applied to classifier-like structures, it was possible to show how information arrows of scattering (dispersion) and information arrows of concentration are identified. Both can be associated with an increase in information entropy in systems that behave like closed ones and can be found in a wide variety of environments. In the ideal version, the Bradford-type distribution is constructed for the visible spectrum of electromagnetic radiation. The corresponding mechanisms are considered in detail using the example of the formation of optical rainbows. Some possibilities are shown for nuclear, atomic, gravitational, quantum and a number of other "rainbows", as well as evolutionary-type structures. The study is interdisciplinary. Within the framework of the presented brief version, we can talk about the junction of primarily information science, environmental science and some areas of physical science.

This volume of “Neutrosophic Sets and Systems” presents a collection of cutting-edge research in the field of neutrosophy and its applications. The papers introduce and explore various extensions of neutrosophic theory, such as neutrosophic spherical cubic sets, which generalize existing fuzzy and interval-valued fuzzy sets. The research also highlights the practical use of these concepts in real-world problem-solving, including project time and cost estimation using triangular neutrosophic PERT analysis and medical diagnosis through the application of neutrosophic linguistic valued hypersoft sets. This collection demonstrates the versatility of neutrosophic methods in handling uncertainty, indeterminacy, and inconsistency across diverse areas, from fuzzy functional analysis to decision-making and optimization problems. The volume underscores the ongoing advancements in neutrosophic theory and its growing potential to provide more robust and nuanced solutions to complex challenges.

This document provides an in-depth exploration of FRACTRAN, an esoteric, Turing-complete programming language invented by John Horton Conway. ? FRACTRAN operates using a list of fractions and a starting integer (input), applying a simple algorithm to produce outputs, often involving prime numbers. ? While computationally inefficient compared to other programming methods, FRACTRAN is valued for its intellectual challenge and mathematical intrigue. ?
Key Points:


Overview: FRACTRAN uses fractions as multipliers to generate sequences of integers. ? Despite requiring many steps to achieve results, it is a fascinating tool for mathematical exploration.


Prime-Game: Conway's "14-fraction prime-producing machine" generates prime numbers through iterative steps. ? Variations include sets of 9, 10, and 14 fractions, each producing sequences of primes. ?


Algorithm: The FRACTRAN algorithm repeatedly multiplies the input by the first fraction that results in an integer product. ? The process halts when no such fraction exists. ?


FRACTRAN ABSTRACT
Applications:

Prime-Game: Produces primes using symbolic or numerical fractions. ?
PI-Game: A flawed 38-fraction program intended to compute decimal digits of ?. ?
Poly-Game: A universal 23-fraction program capable of generating various outputs, though it has unresolved issues. ?



Mathematical Insights: The fractions used in FRACTRAN must be pairwise relatively prime, ensuring unique factorization. ? The primes themselves are not essential; any set of distinct primes can be used. ?


Performance: FRACTRAN is computationally slow. ? For example, Conway's 14-fraction set produces the first 24 primes in 1,000,000 steps, while Kilminster's 10-fraction set generates the first 31 primes in the same number of steps.


Conclusion: FRACTRAN demonstrates the power of simple algorithms to produce complex results. ? Its study can range from superficial appreciation to deep mathematical analysis, revealing intrinsic properties independent of the specific primes used. ?

This document serves as a comprehensive guide to understanding FRACTRAN, its mechanics, and its mathematical significance.

Quantum-mechanical substantiation of a periodic system of isotopes. Models of nuclear orbitals. A.S. Magula Аннотация: Тематика данной статьи лежит в области задач: обоснования периодической системы изотопов и принципа многоуровневой периодичности с помощью квантово-механических расчетов, объединения сильного и электромагнитного взаимодействий и поиска фундаментальной причины периодичности в целом. Данная статья является теоретическим разделом и продолжением статьи: «Периодическая система изотопов», в которой была выполнена проверка системы на соответствие по 10 типам экспериментальных данных, обнаружено периодическое изменение свойств на уровне ядер и вертикальная симметрия подгрупп изотопов. Периодическая система изотопов построена с помощью, специального алгоритмапринципа многоуровневой периодичности атома, от электронов к ядру. В качестве описания многоуровневой периодичности, в данной статье представлена единая система квантовых чисел, с помощью которой выполнено описание, как оболочек электронов, так и нуклонов (биномальная вероятностная интерпретация). С помощью биномальной интерпретации решена задача о частице в одномерной потенциальной яме, выполнен квантово-механический расчет для функций вероятности орбиталей и периодов, как электронов, так и нуклонов,получены характеристические уравнения, воспроизведены проекции электронных орбиталей, показано соответствие биномальной интерпретации семейству сферических гармоник. Для орбиталей электронов выполнен расчет и анализ решений уравнения Шредингера для биномальной интерпретации квантовых чисел. Показана пространственная природа квантовых чисел, для данной интерпретации, в виде степеней свободы. На основе принципа многоуровневой периодичности выведены выражения и выполнены построения плоских проекций орбиталей нуклонов ядра, сделан анализ и выявлено подобие форм с орбиталями электронов. Сделан критический анализ современной сферической системы координат, показаны возможные ошибки в построении орбиталей электронов, с учетом недостатков, предложены две альтернативные сферические системы координат, для которых произведен расчет коэффициентов Ламе и выведены уравнения Лапласа. В качестве поиска фундаментальной причины многоуровневой периодичности, представлена пространственная модель с изменяющимися степенями свободы 0-n, найдено её проявление в природе (формы кристаллов); предложен ряд экспериментов; высказаны прогнозы о применимости принципа многоуровневой периодичности в кварковой теории.

Carnot groups (connected simply connected nilpotent stratified Lie groups) can be endowed with a complex (E * 0 , dc) of "intrinsic" differential forms. In this paper we prove that, in a free Carnot group of step κ, intrinsic 1-forms as well as their intrinsic differentials dc appear naturally as limits of usual "Riemannian" differentials dε, ε > 0. More precisely, we show that L 2 -energies associated with ε -κ dε on 1-forms Γ-converge, as ε → 0, to the energy associated with dc.

I am grateful to my Amath 507 students for their enthusiasm and hard work and for uncovering interesting applications of the calculus of variations. I owe special thanks to William K. Smith for supervising my undergraduate thesis in the calculus of variations (35 years ago) and to Hanno Rund for teaching a fine series of courses on the calculus of variations during my graduate years. Sadly, these two wonderful teachers are now both deceased. I thank the fine editors and reviewers of the American Mathematical Society for their helpful comments. Finally, I thank my family for their encouragement and for putting up with the writing of another book.

We carry out an analysis of the size of the contact surface between a Cheeger set E and its ambient space Ω ⊂ R d . By providing bounds on the Hausdorff dimension of the contact surface ∂E ∩ ∂Ω, we show a fruitful interplay between this size itself and the regularity of the boundaries. Eventually, we obtain sufficient conditions to infer that the contact surface has positive (d -1) dimensional Hausdorff measure. Finally we prove by explicit examples in two dimensions that such bounds are optimal.

Following the global method for relaxation we prove an integral representation result for a large class of variational functionals naturally defined on the space of functions with Bounded Deformation. Mild additional continuity assumptions are required on the functionals.

We prove a sharp quantitative version of Hales' isoperimetric honeycomb theorem by exploiting a quantitative isoperimetric inequality for polygons and an improved convergence theorem for planar bubble clusters. Further applications include the description of isoperimetric tilings of the torus with respect to almost unit-area constraints or with respect to almost flat Riemannian metrics. Nous prouvons une version quantitative optimale du théorème d'isopérimétrie de Hales en exploitant une inégalité isopérimétrique quantitative sur les polygones et une convergence améliorée pour les amas planaires de bulles. Des conséquences incluent la description de pavages isopérimétriques du tore par des contraintes presque unitaires ou par des métriques riemanniennes presque plates.

The aim of this paper is to provide a proof for a version of the Morse inequalities for manifolds with boundary. Our main results are certainly known to the experts on Morse theory, nevertheless it seems necessary to write down a complete proof for it. Our proof is analytic and is based on the J. Roe account of Witten's approach to Morse Theory.

We study the asymptotic growth of the eigenvalues of the Laplace-Beltrami operator on singular Riemannian manifolds, where all geometrical invariants appearing in classical spectral asymptotics are unbounded, and the total volume can be infinite. Under suitable assumptions on the curvature blow-up, we show how the singularity influences the Weyl's asymptotics. Our main motivation comes from the construction of singular Riemannian metrics with prescribed non-classical Weyl's law. Namely, for any non-decreasing slowly varying function υ we construct a singular Riemannian structure whose spectrum is discrete and satisfies Examples of slowly varying functions are log λ, its iterations log k λ = log k-1 log λ, any rational function with positive coefficients of log k λ, and functions with non-logarithmic growth such as exp ((log λ) α1 . . . (log k λ) α k ) for α i ∈ (0, 1). A key tool in our arguments is a new quantitative estimate for the remainder of the heat trace and the Weyl's function on Riemannian manifolds, which is of independent interest.

 denote the Heisenberg group with the usual Carnot-Carathéodory metric d. It is known (since the work of Pansu and Semmes) that the metric space (H, d) cannot be embedded in a bilipschitz fashion into a Hilbert space; however, from a general theorem of Assouad, for any 0 < ε ≤ 1/2, the snowflaked metric space (H, d 1-ε ) embeds into an infinite-dimensional Hilbert space with distortion O(ε -1/2 ). This distortion bound was shown by Austin, Naor, and Tessera to be sharp for the Heisenberg group H. Assouad's argument allows ℓ 2 to be replaced by R D(ε) for some dimension D(ε) dependent on ε. Naor and Neiman showed that D could be taken independent of ε, at the cost of worsening the bound on the distortion to O(ε -1-c D ), where cD → 0 as D → ∞. In this paper we show that one can in fact retain the optimal distortion bound O(ε -1/2 ) and still embed into a bounded dimensional space R D , answering a question of Naor and Neiman. As a corollary, the discrete ball of radius The construction is iterative, and is inspired by the Nash-Moser iteration scheme as used in the isometric embedding problem; this scheme is needed in order to counteract a certain "loss of derivatives" problem in the iteration.

 denote the Heisenberg group with the usual Carnot-Carathéodory metric d. It is known (since the work of Pansu and Semmes) that the metric space (H, d) cannot be embedded in a bilipschitz fashion into a Hilbert space; however, from a general theorem of Assouad, for any 0 < ε ≤ 1/2, the snowflaked metric space (H, d 1-ε ) embeds into an infinite-dimensional Hilbert space with distortion O(ε -1/2 ). This distortion bound was shown by Austin, Naor, and Tessera to be sharp for the Heisenberg group H. Assouad's argument allows ℓ 2 to be replaced by R D(ε) for some dimension D(ε) dependent on ε. Naor and Neiman showed that D could be taken independent of ε, at the cost of worsening the bound on the distortion to O(ε -1-c D ), where cD → 0 as D → ∞. In this paper we show that one can in fact retain the optimal distortion bound O(ε -1/2 ) and still embed into a bounded dimensional space R D , answering a question of Naor and Neiman. As a corollary, the discrete ball of radius The construction is iterative, and is inspired by the Nash-Moser iteration scheme as used in the isometric embedding problem; this scheme is needed in order to counteract a certain "loss of derivatives" problem in the iteration.

 denote the Heisenberg group with the usual Carnot-Carathéodory metric d. It is known (since the work of Pansu and Semmes) that the metric space (H, d) cannot be embedded in a bilipschitz fashion into a Hilbert space; however, from a general theorem of Assouad, for any 0 < ε ≤ 1/2, the snowflaked metric space (H, d 1-ε ) embeds into an infinite-dimensional Hilbert space with distortion O(ε -1/2 ). This distortion bound was shown by Austin, Naor, and Tessera to be sharp for the Heisenberg group H. Assouad's argument allows ℓ 2 to be replaced by R D(ε) for some dimension D(ε) dependent on ε. Naor and Neiman showed that D could be taken independent of ε, at the cost of worsening the bound on the distortion to O(ε -1-c D ), where cD → 0 as D → ∞. In this paper we show that one can in fact retain the optimal distortion bound O(ε -1/2 ) and still embed into a bounded dimensional space R D , answering a question of Naor and Neiman. As a corollary, the discrete ball of radius The construction is iterative, and is inspired by the Nash-Moser iteration scheme as used in the isometric embedding problem; this scheme is needed in order to counteract a certain "loss of derivatives" problem in the iteration.

1 Distance queries are a basic tool in data analysis. They are used for detection and localization of change for the purpose of anomaly detection, monitoring, or planning. Distance queries are particularly useful when data sets such as measurements, snapshots of a system, content, traffic matrices, and activity logs are collected repeatedly. Random sampling, which can be efficiently performed over streamed or distributed data, is an important tool for scalable data analysis. The sample constitutes an extremely flexible summary, which naturally supports domain queries and scalable estimation of statistics, which can be specified after the sample is generated. The effectiveness of a sample as a summary, however, hinges on the estimators we have. We derive novel estimators for estimating Lp distance from sampled data. Our estimators apply with the most common weighted sampling schemes: Poisson Probability Proportional to Size (PPS) and its fixed sample size variants. They also apply when the samples of different data sets are independent or coordinated. Our estimators are admissible (Pareto optimal in terms of variance) and have compelling properties. We study the performance of our Manhattan and Euclidean distance (p = 1, 2) estimators on diverse datasets, demonstrating scalability and accuracy even when a small fraction of the data is sampled. Our work, for the first time, facilitates effective distance estimation over sampled data.

This paper establishes a rigorous bridge between recursive symbolic systems and classical mathematics. At its core is the Φ-field, a formally proven fixed-point attractor that emerges from logical closure, metric contractivity, and adjoint embedding–projection structure. Part One proves the existence, uniqueness, and inevitability of the Φ-field in any system satisfying these axioms, using constructive logic, Banach fixed-point theory, and categorical duality.

Part Two constructs an explicit CSP-compatible projection that embeds recursive dynamics into Banach space operators, preserving convergence, symmetry, and analytic structure. This embedding allows recursive solutions, such as those arising in number theory and operator symmetries, to be represented in a fully verifiable classical mathematical framework. The result is a complete and general projection method for translating recursive fixed-point emergence into CSP-valid mathematics, closing the gap between symbolic recursion and formal analysis.

Two approaches to the simulation of stable and equilibrium longitudinal profiles and slopes are considered.The first one deals with a solution to the equation of solid matter (sediment) continuity in the equilibrium case and the second approach deals with the application of variational principles.The approaches considered and results obtained can be used in hydroengineering construction for designing stable and equilibrium longitudinal profiles of water streams and slopes.

In my humble opinion, we have an unjustified polemic in the world of mathematics, yet again. My background is tertiary level mathematics and concomitant research in specialised areas, so when a friend emailed me the link to this book, I was so excited after reading the author's hype, that I ordered a prepublication copy. My expectations have not been met, unfortunately, hence my analysis precipitated this review.

The theory of uniform distribution of sequences of algebraic integers in a fixed algebraic number field K, as initiated by Kuipers, Niederreiter, and Shiue, is developed from a measuretheoretic viewpoint. After establishing some general facts in § 2, in particular, the analogy between uniform distribution of sequences of algebraic integers in K and of sequences of lattice points, a method of enumerating all algebraic integers in K into a uniformly distributed sequence is discussed in § 3. This enumeration method is useful for the construction of other uniformly distributed sequences as well and plays a role in the density theory. In § 4, a so-called Banach-Buck measure is defined on the ring of all algebraic integers in K. Various relations between this measure and the property of uniform distribution are exhibited. Based on Buck's general concept of density, the notions of relative density and of density of sets of algebraic integers in K are introduced in the final section. Connections among the concepts of uniform distribution, measurability, and relative density of sequences of algebraic integers in K are established. 2* Generalities* Let K be a given algebraic number field of degree [K: Q] = k over the rationale, and let 0 be the ring of all algebraic integers in K. Let I c 0 be a nontrivial integral ideal with norm <yKl. , is a sequence of elements in 0, then A(iV, α + I, J*f) will denote the number of n, 1 ^ n ^ N, such that α Λ = a (mod I). The following two definitions can be found in . DEFINITION 2.1. Let JcO be a nontrivial integral ideal. Then the sequence S/ is uniformly distributed modulo / (u.d. mod I) if for every coset α + I of /. DEFINITION 2.2. The sequence J^ is uniformly distributed in 0 (u.d. in 0) if S/ is u.d. mod I for every nontrivial integral ideal JcO. LEMMA 2.3. Let J£ I be nontrivial integral ideals, and let Sf be a sequence of algebraic integers in 0. Then, if Szf is u.d. mod/, it is also u.d. mod /.

We consider the spectral problem on ∂Ω in a smooth bounded domain Ω of R 2 . The factor ρ ε which appears in the first equation plays the role of a mass density and it is equal to a constant of order ε -1 in an ε-neighborhood of the boundary and to a constant of order ε in the rest of Ω. We study the asymptotic behavior of the eigenvalues λ(ε) and the eigenfunctions u ε as ε tends to zero. We obtain explicit formulas for the first and second terms of the corresponding asymptotic expansions by exploiting the solutions of certain auxiliary boundary value problems.

We show that there are harmonic functions on a ball Bn of R n , n ≥ 2, that are continuous, and even Hölder continuous, up to the boundary but not in the Sobolev space H s (Bn) for s bigger than a certain sharp bound. The idea for the construction is inspired by the two-dimensional example of a harmonic continuous function with infinite energy presented by To obtain examples in any dimension n ≥ 2 we exploit certain series of spherical harmonics.

We show that there are harmonic functions on a ball Bn of R n , n ≥ 2, that are continuous, and even Hölder continuous, up to the boundary but not in the Sobolev space H s (Bn) for s bigger than a certain sharp bound. The idea for the construction is inspired by the two-dimensional example of a harmonic continuous function with infinite energy presented by To obtain examples in any dimension n ≥ 2 we exploit certain series of spherical harmonics.

We consider the spectral problem on ∂Ω in a smooth bounded domain Ω of R 2 . The factor ρ ε which appears in the first equation plays the role of a mass density and it is equal to a constant of order ε -1 in an ε-neighborhood of the boundary and to a constant of order ε in the rest of Ω. We study the asymptotic behavior of the eigenvalues λ(ε) and the eigenfunctions u ε as ε tends to zero. We obtain explicit formulas for the first and second terms of the corresponding asymptotic expansions by exploiting the solutions of certain auxiliary boundary value problems.

We take an open regular domain Ω in R n , n ≥ 3. We introduce a pair of positive parameters ε 1 and ε 2 and we set ε ≡ (ε 1 , ε 2 ). Then we define the perforated domain Ω ε by making in Ω a small hole of size ε 1 ε 2 at distance ε 1 from the boundary. When ε → (0, 0), the hole approaches the boundary while its size shrinks at a faster rate. In Ω ε we consider a Dirichlet problem for the Laplace equation and we denote its solution by u ε . By an approach based on functional analysis and on the introduction of special layer potentials we show that the map which takes ε to (a restriction of) u ε has a real analytic continuation in a neighbourhood of (0, 0).

We introduce numerical invariants of contact forms in dimension three, and use asymptotic cycles to estimate them. As a consequence we prove a version for Anosov Reeb flows of results due to Hutchings and Weiler on mean actions of periodic points. The main tool is the Action-Linking Lemma, expressing the contact area of a surface bounded by periodic orbits as the Liouville average of the asymptotic intersection number of most trajectories with the surface.

Аннотация В статье на основании законов сохранения массы и энергии доказывается ошибочность расчёта энергетического эффекта ядерных реакций на основании «дефекта массы». Предложен вывод соотношения между массой и энергией, отличного от принципа их эквивалентности А. Эйнштейна и исключающего их взаимопревращение. Вводится понятие КПД процесса нуклеосинтеза и даётся новая трактовка процессов синтеза лёгких элементов как аналога «сжигания» ядерного топлива. Вскрывается вопиющее противоречие понятия «дефекта массы» с законом сохранения массы и условиями самопроизвольности процессов. Делается вывод о бесперспективности создания энергоустановок горячего синтеза.