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Through Quantum Lab’s simulations, you can accompany the seven course modules with an exploration of phenomena, circuits and protocols, observing how representations and results change as you adjust the available parameters and returning to the course materials to interpret what you have seen.

GeoStat Lab lets you explore statistical methods used in Earth sciences by comparing samples, time series and spatial data as you change model assumptions and parameters. Visual representations, mathematical explanations and R code accompany each path, helping you interpret the results and reconstruct the analysis that produced them.

When devices, services and identities are described through their relationships, system security can be studied as a structural problem in which both a vulnerability and the paths through which it could expose other resources matter.

Change the parameters and compare the mathematical predictions.

Change the parameters and compare the mathematical predictions.

Change the parameters and compare the mathematical predictions.

The Advanced Calculus and Stochastic Processes entry for 2026/2027 brings together the course references at Università Cattolica del Sacro Cuore, providing access to the institutional page and keeping the information for this academic year distinct from other editions.

This entry concerns the Computer Science component of Mathematics I and Information Technology for Geological Sciences in 2026/2027, with links to the University of Milan’s institutional information about the course.

The seven modules lead from physical foundations and mathematical tools to quantum circuits, algorithms and protocols, with slides available to read and download as PDFs and links in each module to related activities in Quantum Lab.

The Advanced Calculus and Stochastic Processes entry for 2025/2026 brings together the course references at Università Cattolica del Sacro Cuore, providing access to the institutional page and keeping the information for this academic year distinct from other editions.

This entry concerns the Computer Science component of Mathematics I and Information Technology for Geological Sciences in 2025/2026, with links to the University of Milan’s institutional information about the course.

The Advanced Calculus and Stochastic Processes entry for 2024/2025 brings together the course references at Università Cattolica del Sacro Cuore, providing access to the institutional page and keeping the information for this academic year distinct from other editions.

From classical and statistical simulation to waves, interference, two-alternative states and composite systems. The module addresses the loss of interference, information and irreversibility, leading to the transition from quantum mechanics to computation.

Complex numbers, vectors, matrices and tensor products establish the language of computation; probability, states and measurement connect mathematical transformations to observable outcomes.

Reading and executing circuits, single- and multi-qubit gates, composition, phases and interference. Reversible logic leads to the Deutsch and Deutsch–Jozsa algorithms, developed through examples and checks.

Two hidden-string reconstruction problems: the coefficients of a binary parity and the XOR relation between inputs with the same output. Oracles, interference and measurement are followed through to reconstruction of the answer.

Divisors, congruences, inverses and modular powers prepare the construction of RSA. Quantum period finding and the Fourier transform lead to message recovery; a final discussion connects the period to factorisation.

The search for valid candidates is developed through oracles, reflections and rotations in state space. The examples introduce multiple solutions, an unknown number of solutions and amplitude amplification, keeping costs and limitations explicit.

Key distribution with BB84, QKD requirements and limitations, comparison with post-quantum cryptography, superdense coding and entropy. The final part describes quantum codes, syndromes and error recovery.

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Passages and openings connect the surface in all three spatial directions. This form evokes the study of periodic structures and materials whose internal geometry influences their properties.

The surface separates space into two regions connected through periodic openings. Its symmetry and curvature offer a useful reference for understanding three-dimensional networks and porous structures.

The spatial network has connections and openings distinct from those of Schwarz P. Rotating it helps trace the passages and distinguish the regions separated by the surface.

The carbon lattice's hexagons remain visible as the framework curves around its openings. Octagons connect the saddle regions, where the bonds reveal the narrowing and subsequent widening of the structure.

The ideal lattice is displaced out of its plane according to a regular field. One can observe how deformation changes the ribbon’s shape while leaving lattice connectivity recognisable.

Longitudinal curves and circular sections describe three families of necks. The model highlights transitions between regions and the role of curvature in spatial connections.

Variables occupy three lattice-like communities whose amber bridges make the connections between groups visible, while rotation helps distinguish those links from the dependencies within each community.

Program blocks form an architecture of directed paths in which a branch opens three alternatives that later merge. An amber route highlights one permitted sequence, with arrowheads indicating the direction of execution.

The mesh follows a function that maps two variables to a value between zero and one. Three level curves reveal its shape, while the amber line marks the classification threshold, whose curvature depends on the interaction term xy.

A continuous curve combines three values of the same time series at a fixed delay, giving recurring and irregular patterns a spatial form. Denser sampling and a tubular cross-section make the trajectory easier to follow where distinct portions overlap in projection.

A point on the sphere represents a pure state of a qubit, up to a global phase. The highlighted vector selects a state; meridians and parallels help read the two angular coordinates.

Peaks and troughs result from the sum of two periodic contributions. The figure introduces superposition visually, while the second relation shows the cross term responsible for interference between complex amplitudes.

Two lines represent the qubits. A Hadamard transformation on the first, followed by a controlled-X gate, prepares a Bell state from |00⟩. The terminal shapes suggest the circuit’s readout points.

Integer combinations of three vectors generate the lattice points. Highlighted segments distinguish the basis from the point network, providing an initial visual reference for lattice problems.

The curves represent twelve synthetic realisations of geometric Brownian motion. Each path is generated from random increments; a shared component introduces dependence between realisations.

The surface describes a bivariate normal density. Correlation tilts the bell’s cross-sections: observing its shape helps distinguish dispersion from dependence between the two components.

The ellipsoid translates a covariance matrix into spatial directions and magnitudes. The highlighted section and principal axes reveal anisotropy by showing how dispersion varies with direction.

Iterating the same rule produces intricate boundaries between escaping points and points that remain bounded. Colour depends on escape time and reveals details at different scales.

Four scaled-down copies occupy a tetrahedron’s vertices. Repeating the construction produces a structure in which empty space is as recognisably organised as the solids.

Dividing a cube into 27 parts and removing seven generates a structure of interconnected cavities. Recursion reproduces this arrangement at progressively smaller scales.

A half-twist before joining the ends changes the relationship between boundary and surface. Following the strip explores a form in which a local distinction between two sides cannot be extended to a global choice.

A straight line rotates while advancing along an axis, generating a ruled surface. The relationship between rotation and translation reveals the continuity of the form from different viewpoints.

Rotating a catenary generates a surface with a central neck. The contrast between the two principal curvatures makes visible a geometry encountered in the study of minimal surfaces.

Two periodic coordinates traverse the surface in distinct directions. The central hole and the continuity of paths make it a useful form for considering periodicity and topology.

Twelve pentagons allow a hexagonal carbon network to close into a cage, revealing how ring arrangement accompanies the curvature of a closed carbon surface.

Cutting a graphene sheet leaves an atomic edge whose properties can be investigated through spectroscopy and calculations. The two views make armchair and zigzag terminations recognisable.

Twisting moves the hexagons out of plane and changes the orientation of ribbon regions, providing a visual introduction to the relationship between structural distortion and electronic properties.

Two overlapping graphene fragments reveal how a relative shift changes ring alignment, introducing the role of interlayer interactions in the study of optical properties.

Alerts in European schools show how an unfounded threat can consume resources, change decisions and prolong its effects. This essay examines that pressure through cognitive overload, orientation and the OODA loop in John Boyd’s thinking.

Starting from altered commands and false alarms, the essay follows the checks, decisions and shifts of resources that a signal can provoke, drawing on Liddell Hart and John Boyd to examine how disrupted information affects an organisation’s freedom of action and cohesion.

Because models that correctly classify the same programs may rely on different cues, the work examines the stability of explanations in behavioural-trace classification, considering data representations and the regularities learned by classifiers.

Through a survey of 72 municipalities and a prototype for classifying municipal website pages, the volume examines uses of artificial intelligence in local government in relation to the skills and responsibilities accompanying its adoption.

Traces left by programs in processor activity are used to recognise suspicious behaviour through a methodology that connects ransomware detection with an examination of the relationships between variables underlying classification.

The study examines how prepared Lombardy’s municipalities are to face cyber threats, identifying support and training needs that affect the conditions in which local administrations can address security problems.

Asteroid classification provides a setting in which physical descriptions of the phenomenon make it possible to examine whether the relationships used by an algorithm to make predictions are consistent with the knowledge used to interpret the data.

By comparing what a model appears to learn from data with known physical laws, the work examines the relationship between prediction and explanation and the meaning that can be attributed to the relationships underlying an algorithm’s responses.

The study examines atomic vibrations in graphene nanoribbons, through which a signature of edge structure can be obtained and the shape of the nanostructure related to its vibrational properties.

The work studies how controlled deformation of a graphene nanoribbon can change the energy required to excite its electrons, relating changes in shape to the resulting variations in optical properties.

Through light scattered by graphene nanoribbons, the work examines information about their width, edge structure and attached chemical groups, relating these features to the material’s Raman signature.

The work compares the optical properties of small stacked graphene fragments with those of isolated fragments, examining how layer alignment and edge chemistry contribute to differences in light absorption.

The arrangement of bonds between atoms is described through mathematical indices that summarise aspects of connectivity, allowing different carbon structures to be compared through quantities computed on the graph.

By representing atoms as nodes and bonds as connections, the work relates some physical properties of carbon surfaces to simpler topological descriptions through which differently organised structures can be compared.

The chapter proposes two possible crystalline arrangements of carbon and studies their structure and properties theoretically, using computational design to examine new cage configurations.

Closing a nanoribbon into a Möbius form changes its structural connections, which the work describes through topological indices that quantify differences introduced by the twist and allow their organisation to be compared.

The study adds information about the reactivity of atoms and bonds to a molecular network’s description of atomic connections, allowing the representation of connectivity to include aspects of chemical characterisation.

Epioptics-11 · World Scientific, pp. 114–131

The chapter traces the relationship between negatively curved carbon networks, the conditions in which they can grow and their physical properties, examining how schwarzite geometry contributes to the description of porous carbon.

How can we evaluate the evidence on which a classification is based?

Which execution traces can identify malicious behaviour?

Which properties of a nanostructure depend on its geometry and bond arrangement?

How can a forecast be evaluated when data change over time?

What warrants treating a prediction as an explanation?

Which problems offer an advantage in the quantum model of computation?

How can the operation of computing systems be explained in a verifiable way?