Differential Topology

This paper proposes a new differential topology that features a stacked multiloop inductor. Comparative studies of stacked one-to four-loop spiral inductors with and without patterned ground shields (PGSs) for silicon-based radio-frequency integrated circuits (RFICs) were conducted, and lumped-element circuit models were developed for these inductors. The partialelement equivalent-circuit method that can accurately analyze mutual inductive couplings among different spirals in these multiloop geometries was employed for capturing the frequencydependent inductances and resistances of inductors at low frequencies. A good agreement between numerical results and measurements is obtained. It is demonstrated that a stacked multiloop spiral inductor with differential topology and PGS has a larger inductance and a higher Q-factor as compared with the same inductor without differential topology and PGS. This hybrid methodology could provide a promising technique for developing new silicon-based passive devices used in RFICs.

This paper introduces a new class of geometric structure, termed SynchronoGeometry, which integrates localized temporal rhythms into the spatial configuration of manifolds. Unlike conventional geometries that treat time as an external parameter, SynchronoGeometry assigns an intrinsic time value to each point in space, resulting in dynamic metrics and temporally modulated geodesics. The proposed framework paves the way for modeling systems where localized time variation influences curvature and form-offering potential applications in theoretical physics, interactive media design, and cognitive simulation.

Основная трудность большинства современной исследовательской
литературы по эпистемологии математики заключается в том, что она не рассматривает современную математическую практику. Современная практика - которую мы будем условно считать развитием событий с 1940-х годов по настоящее время - возвестила о беспрецедентном распространении математических знаний, применимых к физике и точным наукам.

Математические структуры, возникающие в процессе математизации, таким образом, проходят самоотбор на предмет эмпирической и математической полезности. Окончательное развитие структур Калаби-Яу ante-rem является результатом интуитивного диалога с физическими науками и приводит к целенаправленно общей классифицирующей (аристотелизирующей) онтологии. С точки зрения математики, многообразия Калаби-Яу представляют аристотелевскую (контекстуализирующую) тенденцию; полученные структуры, многообразия Калера и Калера-Эйнштейна, представляют платонизирующую тенденцию, вызванную релятивистским поиском анте-ремовской структуры пространства-времени.

Đây là tập bài giảng cho các môn học về tôpô cho sinh viên đại học, gồm Tôpô Đại cương, Tôpô Đại số, và Tôpô Vi phân.

This is a set of lecture notes for courses in topology for undergraduate students, consisting of General Topology, Algebraic Topology, and Differential Topology.

The de Rham theorem gives a natural isomorphism between De Rham cohomology and singular cohomology on a paracompact differentiable manifold. We proved this theorem on a wider family of subsets of Euclidean space, on which we can define inner differentiability. Here we define this family of sets called tangentially locally linearly independent sets, propose inner differentiability on them, postulate usual properties of differentiable real functions and show that the integration over sets that are wider than manifolds is possible.

The problem of the existence of non-medial distributive hamiltonian quasigroups is solved. Translating this problem first to commutative Moufang loops with operators, then to ternary algebras and, finally, to cocyclic modules over\linebreak $\Bbb Z[x,x^{-1},(1-x)^{-1}]$, it is shown that every non-medial distributive hamiltonian quasigroup has at least 729 elements and that there are just two isomorphism classes of such quasigroups of the least cardinality. The quasigroups representing these two classes are anti-isomorphic.

Programa de doctorat de Matem atica Aplicada i An alisi. Bienni 92-94. Mem oria presentada pera aspirar al grau de Doctor en Ci encies Matem atiques perla Universitat de Barcelona. Certi co que la present mem oria ha estat realitzada per n' Alex Haro i Provinciale, i dirigida per mi. Barcelona, 3 d e juliol de 1998. Carles Sim o i Torres A la Cristina ' 1:839286755: We shall refer to it as the golden vector. Its rst convergents are

TAKEUTIIZARING. Introduction to 34 SPITZER. Principles of Random Walk. Axiomatic Set Theory. 2nd ed. 2nd ed. 2 OXTOBY. Measure and Category. 2nd ed. 35 ALEXANDERIWERMER. Several Complex 3 SCHAEFER. Topological Vector Spaces. Variables and Banach Algebras. 3rd ed. 2nd ed. 36 KELLEy/NAMIOKA et al. Linear 4 HILTON/STAMMBACH. A Course in Topological Spaces. Homological Algebra. 2nd ed. 37 MONK. Mathematical Logic. 5 MAC LANE. Categories for the Working 38 GRAUERTIFRITZSCHE. Several Complex Mathematician. 2nd ed. Variables. 6 HUGHESIPIPER. Projective Planes. 39 ARVESON. An Invitation to C*-Algebras. 7 J.-P. SERRE. A Course in Arithmetic. 40 KEMENY/SNELIiKNAPP. Denumerable 8 TAKEUTIIZARING. Axiomatic Set Theory. Markov Chains. 2nd ed. 9 HUMPHREYS. Introduction to Lie Algebras 41 APOSTOL. Modular Functions and and Representation Theory. Dirichlet Series in Number Theory. 10 COHEN. A Course in Simple Homotopy 2nd ed. Theory. 42 J.-P. SERRE. Linear Representations of 11 CONWAY. Functions of One Complex Finite Groups. Variable I. 2nd ed. 43 GILLMAN/JERISON. Rings of Continuous 12 BEALs. Advanced Mathematical Analysis. Functions. 13 ANDERSONIFULLER. Rings and Categories 44 KENDIG. Elementary Algebraic Geometry. of Modules. 2nd ed. 45 LoilVE. Probability Theory I. 4th ed. 14 GOLUBITSKy/GUILLEMIN. Stable Mappings 46 LoilVE. Probability Theory II. 4th ed. and Their Singularities. 47 MOISE. Geometric Topology in 15 BERBERIAN. Lectures in Functional Dimensions 2 and 3. Analysis and Operator Theory. 48 SAcHSiWu. General Relativity for 16 WINTER. The Structure of Fields. Mathematicians. 17 ROSENBLATT. Random Processes. 2nd ed. 49 GRUENBERGIWEIR. Linear Geometry. 18 HALMos. Measure Theory. 2nd ed. 19 HALMos. A Hilbert Space Problem Book. 50 EDWARDS. Fermat's Last Theorem. 2nd ed. 51 KLINGENBERG. A Course in Differential 20 HUSEMOLLER. Fibre Bundles. 3rd ed. Geometry. 21 HUMPHREYS. Linear Algebraic Groups. 52 HARTSHORNE. Algebraic Geometry. 22 BARNESIMACK. An Algebraic Introduction 53 MANIN. A Course in Mathematical Logic. to Mathematical Logic. 54 GRAVERIWATKINS. Combinatorics with 23 GREUB. Linear Algebra. 4th ed. Emphasis on the Theory of Graphs. 24 HOLMES. Geometric Functional Analysis 55 BROWNIPEARCY. Introduction to Operator and Its Applications. Theory I: Elements of Functional Analysis. 25 HEWITT/STROMBERG. Real and Abstract 56 MASSEY. Algebraic Topology: An Analysis. Introduction. 26 MANES. Algebraic Theories. 57 CROWELL!FOX. Introduction to Knot 27 KELLEY. General Topology. Theory. 28 ZARisKiiSAMUEL. Commutative Algebra. 58 KOBLITZ. p-adic Numbers, p-adic Vol.I. Analysis, and Zeta-Functions. 2nd ed. 29 ZARisKiiSAMUEL. Commutative Algebra. 59 LANG. Cyclotomic Fields. VaLlI. 60 ARNOLD. Mathematical Methods in 30 JACOBSON. Lectures in Abstract Algebra I. Classical Mechanics. 2nd ed. Basic Concepts.

The investigation of the Hamiltonian dynamical counterpart of phase transitions, combined with the Riemannian geometrization of Hamiltonian dynamics, has led to a preliminary formulation of a differential-topological theory of phase transitions. In fact, in correspondence of a phase transition there are peculiar geometrical changes of the mechanical manifolds that are found to stem from changes of their topology. These findings, together with two theorems, have suggested that a topological theory of phase transitions can be formulated to go beyond the limits of the existing theories. Among other advantages, the new theory applies to phase transitions in smallNsystems (that is, at nanoscopic and mesoscopic scales), and in the absence of symmetry-breaking. However, the preliminary version of the theory was incomplete and still falsifiable by counterexamples. The present work provides a relevant leap forward leading to an accomplished development of the topological theory of phase tran...

The projective span of a smooth manifold is defined to be the maximal number of linearly independent tangent line fields. We initiate a study of projective span, highlighting its relationship with the span, a more classical invariant. We calculate the projective span for all Wall manifolds, which are certain mapping tori of Dold manifolds.

En (1) se prueba que un espacio de Banach E verifica la propiedad de aproximación de Grothendieck si y sólo si el espacio 9í (E) de las funciones holomorfas sobre E con la topología compacto abierta, la verifica. El objeto del presente trabajo es establecer un resultado análogo para el caso de funciones diferenciables reales. Los intentos realizados en esta dirección han tenido, hasta ahora, un éxito parcial (véase (2)). En los trabajos anteriormente citados, se utiliza de forma esencial la caracterización de la propiedad de aproximación en términosdel e-producto. Desde nuestro punto de vista, las dificultades radican principalmente en que la topología usual en los espacios C" (E, F) de la convergencia uniforme de las funciones y sus derivadas sobre los compactos de E, no parece la más adecuada para, en primer lugar, establecer la igualdad C" 1 (E) € F = C n (E, F), y en segundo lugar para estudiar la densidad de C™ (E) (g) F en C n (E, F). Por esta última razón, en (5) se introducen en los espacios C" (E, F) otras topologías, que permiten obtener resultados satisfactorios en la dirección citada. Por otro lado, debido a las dificultades que presenta la diferencial de Fréchet para estudiar propiedades de completitud en los es-1. La diferencial compacta. Propiedades de densidad

Topological integrals appear frequently in Lagrangian field theories. On manifolds without boundary, they can be treated in the framework of (absolute) (co)homology using the formalism of Cheeger-Simons differential characters. String and D-brane theory involve field theoretic models on worldvolumes with boundary. On manifolds with boundary, the proper treatment of topological integrals requires a generalization of the usual differential topological set up and leads naturally to relative (co)homology and relative Cheeger-Simons differential characters. In this paper, we present a construction of relative Cheeger-Simons differential characters which is computable in principle and which contains the ordinary Cheeger-Simons differential characters as a particular case.

We prove that every family of (not necessarily distinct) odd cycles O 1 , . . . , O 2⌈n/2⌉-1 in the complete graph K n on n vertices has a rainbow odd cycle (that is, a set of edges from distinct O i 's, forming an odd cycle). As part of the proof, we characterize those families of n odd cycles in K n+1 that do not have any rainbow odd cycle. We also characterize those families of n cycles in K n+1 , as well as those of n edge-disjoint nonempty subgraphs of K n+1 , without any rainbow cycle.

En este trabajo se presenta el problema de clasificación de las foliaciones del plano y el problema de representación de una foliación del plano mediante una recta y semirrectas. El primer problema, se estudiará a través de la clasificación : 1. De las variables topológicas de dimensión uno, con base enumerable y simplemente conexa. 2. De los fibrados sobre tal variedad, localmente triviales con espacio total de Hausdorff. Luego se clasifican las estructuras diferenciables sobre la ramificación simple y se estudian las realizaciones de tales estructuras mediante foliaciones del plano. Para el segundo problema se da una construcción llamada representación y se introduce su correspondiente topología

It is known that the kinematics on the Lorentzian surfaces changes according to the casual characters of the vector fields. Suspicions, the character of the generator curve affects the surface growth. Therefore, we determine the model of the growth function in the three‐dimensional Minkowski spacetime with a null generating curve. Moreover, the proposed method is illustrated with various examples.

It is known that neither immersions nor maps with a fixed finite set of multisingularities are enough to realize all mod 2 homology classes in manifolds. In this paper we define the notion of realizing a homology class up to cobordism; it is shown that for realization in this weaker sense immersions are sufficient, but maps with a fixed finite set of multisingularities are still insufficient.

We prove that every family of (not necessarily distinct) odd cycles O 1 ,. .. , O 2⌈n/2⌉−1 in the complete graph K n on n vertices has a rainbow odd cycle (that is, a set of edges from distinct O i 's, forming an odd cycle). As part of the proof, we characterize those families of n odd cycles in K n+1 that do not have any rainbow odd cycle. We also characterize those families of n cycles in K n+1 , as well as those of n edge-disjoint nonempty subgraphs of K n+1 , without any rainbow cycle.

Given a finite and connected two-dimensional CW-complex K with fundamental group Π and second integer cohomology group H 2 (K; Z) finite of odd order, we prove that: (1) for each local integer coefficient system α : Π → Aut(Z) over K, the corresponding twisted cohomology group H 2 (K; α Z) is finite of odd order, we say order c * (α), and there exists a natural function-which resemble that one defined by the twisted degree-from the set [K; RP 2 ] * α of the based homotopy classes of based maps inducing α on π 1 into H 2 (K; α Z), which is a bijection; (2) the set [K; RP 2 ] α of the (free) homotopy classes of based maps inducing α on π 1 is finite of order c(α) = (c * (α) + 1)/2; (3) all but one of the homotopy classes [f ] ∈ [K; RP 2 ] α are strongly surjective, and they are characterized by the non-nullity of the induced homomorphism f * : H 2 (RP 2 ; ̺ Z) → H 2 (K; α Z), where ̺ is the nontrivial local integer coefficient system over the projective plane. Also some calculations of the groups H 2 (K; α Z) are provided for several two-complexes K and actions α, allowing to compare H 2 (K; Z) and H 2 (K; α Z) for nontrivial α.

Let G be a simple connected graph. Then χ L (G) and χ D (G) will denote the locating chromatic number and the distinguishing chromatic number of G, respectively. In this paper, we investigate a comparison between χ L (G) and χ D (G). In fact, we prove that χ D (G) ≤ χ L (G). Moreover, we determine some types of graphs whose locating and distinguishing chromatic numbers are equal. Specially, we characteristic all graph G with property that χ D (G) = χ L (G) = 3.

The de Rham theorem gives a natural isomorphism between De Rham cohomology and singular cohomology on a paracompact differentiable manifold. We proved this theorem on a wider family of subsets of Euclidean space, on which we can define inner differentiability. Here we define this family of sets called tangentially locally linearly independent sets, propose inner differentiability on them, postulate usual properties of differentiable real functions and show that the integration over sets that are wider than manifolds is possible.

Orientador: Lucas Catão de Freitas FerreiraDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação CientíficaResumo: Estudamos as equações de Navier-Stokes (NS) em...Observação: O resumo, na íntegra, poderá ser visualizado no texto completo da tese digitalAbstract: We study the Navier-Stokes equations (NS) in...Note: The complete abstract is available with the full electronic documentMestradoMatematicaMestre em Matemátic

Topological integrals appear frequently in Lagrangian field theories. On manifolds without boundary, they can be treated in the framework of (absolute) (co)homology using the formalism of Cheeger-Simons differential characters. String and D-brane theory involve field theoretic models on worldvolumes with boundary. On manifolds with boundary, the proper treatment of topological integrals requires a generalization of the usual differential topological set up and leads naturally to relative (co)homology and relative Cheeger-Simons differential characters. In this paper, we present a construction of relative Cheeger-Simons differential characters which is computable in principle and which contains the ordinary Cheeger-Simons differential characters as a particular case.

We show that the global aspects of Abelian and center projection of a SU(2) gauge theory on an arbitrary manifold are naturally described in terms of smooth Deligne cohomology. This is achieved through the introduction of a novel type of differential topological structure, called Cho structure. Half integral monopole charges appear naturally in this framework.

The metric dimension dim(G) of a graph G is the minimum number of vertices such that every vertex of G is uniquely determined by its vector of distances to the chosen vertices. The zero forcing number Z(G) of a graph G is the minimum cardinality of a set S of black vertices (whereas vertices in V (G)\S are colored white) such that V (G) is turned black after finitely many applications of "the color-change rule": a white vertex is converted black if it is the only white neighbor of a black vertex. We show that dim(T) ≤ Z(T) for a tree T , and that dim(G) ≤ Z(G) + 1 if G is a unicyclic graph; along the way, we characterize trees T attaining dim(T) = Z(T). For a general graph G, we introduce the "cycle rank conjecture". We conclude with a proof of dim(T) − 2 ≤ dim(T + e) ≤ dim(T) + 1 for e ∈ E(T).

En el presente trabajo se demuestra que la topologla se ud om et r-Lc a sobre un conjunto X' obtenida a partir de una funcion 6:X~ffi, coincide con la topologla inicial sobre X inducida por la misma funcion; se caracterizan sus conjuntos compactos y se demuestra que cuando 6 es sobreyectiva, la topologla usual de ffi es la cociente por la relaci6n que identifica puntos de X que poseen lamisma imagen per 6• Sed emu est r a a de mas 'q u e 1a top 010 g i a 'e obre .'. los reales no-estandar finitos ffi F , es la topo-* .~. * Est:lR r-+ lR, f u n c i o n que a so c r a a .cada a de lR'f-' apo:rtes su parte estandar. Con esto y los resultados at eriores, se obtienen demostraciones sencillas de algunas caracterizaciones en [3].

En el presente trabajo se demuestra que la topologla se ud om et r-Lc a sobre un conjunto X' obtenida a partir de una funcion 6:X~ffi, coincide con la topologla inicial sobre X inducida por la misma funcion; se caracterizan sus conjuntos compactos y se demuestra que cuando 6 es sobreyectiva, la topologla usual de ffi es la cociente por la relaci6n que identifica puntos de X que poseen lamisma imagen per 6• Sed emu est r a a de mas 'q u e 1a top 010 g i a 'e obre .'. los reales no-estandar finitos ffi F , es la topo-* .~. * Est:lR r-+ lR, f u n c i o n que a so c r a a .cada a de lR'f-' apo:rtes su parte estandar. Con esto y los resultados at eriores, se obtienen demostraciones sencillas de algunas caracterizaciones en [3].

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2019, Director: Carlos Currás Bosch[en] This paper deals with a classic theorem in differential geometry of surfaces: Bonnet’s theorem. Our objective is to generalize the theorem for manifolds of arbitrary codimension. The theory of vector bundles gives us strong tools concerning different topics in differentialgeometry theory, such as metric, connections and curvature. Henceforth, we develop in the first part of this work the most important aspects in this theory, without loosing sight of our objectives. We will introduce one of the most powerful tools in this theory: the pullback of vector bundles. This tool will be frequently used during this work. Moreover, we will shortly introduce the concept of Ehresmann connection, whose presence in modern differential geometry is highly important. At the end, it will be given a precise statement and a proof of the Bonnet’s immersion theorem

Given a finite and connected two-dimensional CW-complex K with fundamental group Π and second integer cohomology group H 2 (K; Z) finite of odd order, we prove that: (1) for each local integer coefficient system α : Π → Aut(Z) over K, the corresponding twisted cohomology group H 2 (K; α Z) is finite of odd order, we say order c * (α), and there exists a natural function-which resemble that one defined by the twisted degree-from the set [K; RP 2 ] * α of the based homotopy classes of based maps inducing α on π 1 into H 2 (K; α Z), which is a bijection; (2) the set [K; RP 2 ] α of the (free) homotopy classes of based maps inducing α on π 1 is finite of order c(α) = (c * (α) + 1)/2; (3) all but one of the homotopy classes [f ] ∈ [K; RP 2 ] α are strongly surjective, and they are characterized by the non-nullity of the induced homomorphism f * : H 2 (RP 2 ; ̺ Z) → H 2 (K; α Z), where ̺ is the nontrivial local integer coefficient system over the projective plane. Also some calculations of the groups H 2 (K; α Z) are provided for several two-complexes K and actions α, allowing to compare H 2 (K; Z) and H 2 (K; α Z) for nontrivial α.

Let H be the class of bounded measurable symmetric functions on [0, 1] 2. For a function h ∈ H and a graph G with vertex set {v 1 ,. .. , v n } and edge set E(G), define t G (h) = • • • {vi,vj }∈E(G) h(x i , x j) dx 1 • • • dx n. We prove that if t G (|h|) 1/|E(G)| is a norm on H, then G is edgetransitive.

Reçu le 12 septembre 2002 ; accepté après révision le 8 octobre 2002 Note présentée par Michèle Vergne. Résumé Nous étudions l'espace Pl(c, λ) des plongements symplectiques de la boule fermée B 4 (c) ⊂ R 4 de capacité c dans (S 2 × S 2 , (1 + λ)ω st ⊕ ω st). Lorsque λ = 0, nous montrons que cet espace se comporte comme les plongements différentiables ordinaires et a donc un type d'homotopie indépendant de la valeur de c ; nous montrons ensuite que si λ > 0 l'application de restriction Pl(c , λ) → Pl(c, λ) cesse d'être une équivalence d'homotopie quand c et c se trouvent de part et d'autre de la valeur λ.

This paper deals with the compensation of analog imperfections in a Ka-Band FMCW SAR. Due to the presence of phase distortion in the up-conversion and down-conversion block, we demonstrate that the calibration of the VCO based on a reference beat signal is range-limited. We propose a post-processing method to compensate the residual sinusoidal nonlinearities of the VCO characteristic as well as the phase distortion coming from the upconversion and down-conversion block. Processing of SAR data acquisition demonstrates the efficiency of the method.

Catastrophe theory is a recently developed branch of topology which has a number of practical applications, principally because o f i t s ability to model situations which include discontinuities or singularities, where the methods of diff'erential calculus break down. Some examples of its possible use in geology are presented, in fields as diverse as volcanology, sedimentology, and structural geology, and wider appfieations, in modeling spatial or temporal discontinuities in other fields, are suggested.

It is shown, via a number of examples in linear algebra and control, optimization, and geometry, that the transversality theorem of differential topology is a useful tool for establishing genericity of a property which is a function of a finite number of real parameters.

This paper presents CMOS front-end ICs with 13.3 dBm output power for K-band FMCW radar, which is integrated in 0.13-μm CMOS technology. The transmitter consists of a voltage controlled oscillator, divider chain, power amplifier, and additional buffers. The receiver consists of a low-noise amplifier, IQ mixers, an IQ generator, and buffers. The leakage problem can be mitigated by adopting differential topology and ground shielding. As a result, the receiver achieves a conversion gain of 35.7 dB, a P1dB of-31.6 dBm, and a DSB noise figure of 5.5 dB. The transmitter achieves the tuning range of 23.8~24.5 GHz and the phase noise of-104 dBc/Hz @ 1MHz offset. The receiver and transmitter chips consume 121.5 mW and 373.5 mW from a 1.5 V power supply, respectively. Using these two chips, the K-band FMCW radar module is implemented and verified by measuring the distance of an object.

Witten constructed topological invariants of manifolds by introducing actions depending only on smooth structures of the manifold without using a metric. He focused on an example of Chern-Simons' theory on 3D manifolds. Sławianowski based his field theory on frame/coframe fields without a metric. A Witten-type topological invariant can be introduced in Sławianowski's framework, using an integral over the reper fields. Such invariant behaves differently for group manifolds than for all others. It additionally distinguishes semi-simple Lie group manifolds from other group manifolds. Further study of this new invariant is needed, including possible generalizations to other field theories.

The investigation of the Hamiltonian dynamical counterpart of phase transitions, combined with the Riemannian geometrization of Hamiltonian dynamics, has led to a preliminary formulation of a differential-topological theory of phase transitions. In fact, in correspondence of a phase transition there are peculiar geometrical changes of the mechanical manifolds that are found to stem from changes of their topology. These findings, together with two theorems, have suggested that a topological theory of phase transitions can be formulated to go beyond the limits of the existing theories. Among other advantages, the new theory applies to phase transitions in smallNsystems (that is, at nanoscopic and mesoscopic scales), and in the absence of symmetry-breaking. However, the preliminary version of the theory was incomplete and still falsifiable by counterexamples. The present work provides a relevant leap forward leading to an accomplished development of the topological theory of phase tran...

This article presents development of a UAV based frequency modulated continuous wave (FMCW) radar system for remotely sensing the water contained within snowpacks. To make the radar system compatible with the payload requirements of small UAV platforms, the radar electronics are implemented with CMOS technology, and the antenna is implemented as an extremely compact and lightweight metasurface (MTS) antenna. This article will discuss how the high absorption losses of snowpacks lead to dynamic range requirements much stricter than FMCW radars used for automotive and other sensing applications, and how these requirements are met through antenna isolation, leakage calibration and exploitation of the range correlation effect. The article discusses in detail the implementation of the radar system, the CMOS microwave and digital circuitry, and the MTS antenna. The developed radar was mounted on a drone and conducted surveys in both Idaho and Alaska during the 2022-2023 winter season. We present several of those field results.

This is a self contained set of lecture notes. The notes were written by Rob van der Vorst. The solution manual is written by Guit-Jan Ridderbos. We follow the book 'Introduction to Smooth Manifolds' by John M. Lee as a reference text [1]. Additional reading and exercises are take from 'An introduction to manifolds' by Loring W. Tu [2].

This article presents development of a UAV based frequency modulated continuous wave (FMCW) radar system for remotely sensing the water contained within snowpacks. To make the radar system compatible with the payload requirements of small UAV platforms, the radar electronics are implemented with CMOS technology, and the antenna is implemented as an extremely compact and lightweight metasurface (MTS) antenna. This article will discuss how the high absorption losses of snowpacks lead to dynamic range requirements much stricter than FMCW radars used for automotive and other sensing applications, and how these requirements are met through antenna isolation, leakage calibration and exploitation of the range correlation effect. The article discusses in detail the implementation of the radar system, the CMOS microwave and digital circuitry, and the MTS antenna. The developed radar was mounted on a drone and conducted surveys in both Idaho and Alaska during the 2022-2023 winter season. We present several of those field results.

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This note gives a syntactic presentation for partial algebraic theories (see [1] and [3]). The logic, called left exact logic, is interpretable in any category with all finite limits, and it has coherent logic as a conservative extension, which implies a completeness theorem.

This article presents development of a UAV based frequency modulated continuous wave (FMCW) radar system for remotely sensing the water contained within snowpacks. To make the radar system compatible with the payload requirements of small UAV platforms, the radar electronics are implemented with CMOS technology, and the antenna is implemented as an extremely compact and lightweight metasurface (MTS) antenna. This article will discuss how the high absorption losses of snowpacks lead to dynamic range requirements much stricter than FMCW radars used for automotive and other sensing applications, and how these requirements are met through antenna isolation, leakage calibration and exploitation of the range correlation effect. The article discusses in detail the implementation of the radar system, the CMOS microwave and digital circuitry, and the MTS antenna. The developed radar was mounted on a drone and conducted surveys in both Idaho and Alaska during the 2022-2023 winter season. We present several of those field results.

The problem of the existence of non-medial distributive hamiltonian quasigroups is solved. Translating this problem first to commutative Moufang loops with operators, then to ternary algebras and, finally, to cocyclic modules over Z[x, x −1 , (1−x) −1 ], it is shown that every non-medial distributive hamiltonian quasigroup has at least 729 elements and that there are just two isomorphism classes of such quasigroups of the least cardinality. The quasigroups representing these two classes are anti-isomorphic. 1991 Mathematics Subject Classification. 20N05.